Derivative function：
Enter an original function (that is, the function to be derived), then set the variable to be derived and the order of the derivative, and click the "Next" button to obtain the derivative function of the corresponding order of the function.
Note that the input function supports mathematical functions and other constants.
Enter an original function (that is, the function to be derived), then set the variable to be derived and the order of the derivative, and click the "Next" button to obtain the derivative function of the corresponding order of the function.
Note that the input function supports mathematical functions and other constants.
Current location：Derivative function > Derivative function calculation history > Answer
There are 1 questions in this calculation: for each question, the 2 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ second\ derivative\ of\ function\ ln(sqrt({(\frac{1}{sin(2x)} - \frac{cos(2x)}{sin(2x)})}^{2}))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = ln(sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)}))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ln(sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)}))\right)}{dx}\\=&\frac{(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})*\frac{1}{2}}{(sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)}))(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}}\\=&\frac{-2cos^{3}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{4cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{2}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2cos^{3}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{4cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{2}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})}\right)}{dx}\\=&\frac{-2(\frac{\frac{-1}{2}(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}})cos^{3}(2x)}{sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2*-3cos(2x)*2cos^{3}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{4}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2*-3cos^{2}(2x)sin(2x)*2}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2cos^{3}(2x)*-(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})*\frac{1}{2}}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}} - \frac{2(\frac{\frac{-1}{2}(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}})cos(2x)}{sin(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2*-cos(2x)*2cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{2}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2*-sin(2x)*2}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2cos(2x)*-(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})*\frac{1}{2}}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin(2x)(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}} + \frac{4(\frac{\frac{-1}{2}(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}})cos^{2}(2x)}{sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{4*-3cos(2x)*2cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{4}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{4*-2cos(2x)sin(2x)*2}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{4cos^{2}(2x)*-(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})*\frac{1}{2}}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}} + \frac{2(\frac{\frac{-1}{2}(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}})}{sin(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{2*-cos(2x)*2}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{2}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{2*-(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})*\frac{1}{2}}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin(2x)(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}} - \frac{2(\frac{\frac{-1}{2}(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}})cos(2x)}{sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2*-3cos(2x)*2cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{4}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2*-sin(2x)*2}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2cos(2x)*-(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})*\frac{1}{2}}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}}\\=&\frac{-4cos^{6}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{6}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{8cos^{4}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{4}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{16cos^{5}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{6}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{24cos^{3}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{4}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{24cos^{4}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{6}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{12cos^{4}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{4}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{16cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{2}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{4cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{2}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{8cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{2}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{24cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{4}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{16cos^{3}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{6}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{24cos^{3}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{4}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{20cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{2}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{8cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{4}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{4}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{4cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{6}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{12cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{4}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{24cos^{4}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{6}(2x)} - \frac{4cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{2}(2x)} - \frac{4cos^{6}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{6}(2x)} - \frac{8cos^{4}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{4}(2x)} + \frac{16cos^{3}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{6}(2x)} - \frac{4}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{2}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{16cos^{5}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{6}(2x)} + \frac{8cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{2}(2x)} + \frac{24cos^{3}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{4}(2x)} - \frac{24cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{4}(2x)} + \frac{4}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{2}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{8cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{4}(2x)} - \frac{4cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{6}(2x)} - \frac{4}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{2}(2x)}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ second\ derivative\ of\ function\ ln(sqrt({(\frac{1}{sin(2x)} - \frac{cos(2x)}{sin(2x)})}^{2}))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = ln(sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)}))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ln(sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)}))\right)}{dx}\\=&\frac{(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})*\frac{1}{2}}{(sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)}))(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}}\\=&\frac{-2cos^{3}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{4cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{2}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2cos^{3}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{4cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{2}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})}\right)}{dx}\\=&\frac{-2(\frac{\frac{-1}{2}(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}})cos^{3}(2x)}{sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2*-3cos(2x)*2cos^{3}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{4}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2*-3cos^{2}(2x)sin(2x)*2}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2cos^{3}(2x)*-(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})*\frac{1}{2}}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}} - \frac{2(\frac{\frac{-1}{2}(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}})cos(2x)}{sin(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2*-cos(2x)*2cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{2}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2*-sin(2x)*2}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2cos(2x)*-(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})*\frac{1}{2}}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin(2x)(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}} + \frac{4(\frac{\frac{-1}{2}(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}})cos^{2}(2x)}{sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{4*-3cos(2x)*2cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{4}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{4*-2cos(2x)sin(2x)*2}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{4cos^{2}(2x)*-(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})*\frac{1}{2}}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}} + \frac{2(\frac{\frac{-1}{2}(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}})}{sin(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{2*-cos(2x)*2}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{2}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{2*-(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})*\frac{1}{2}}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin(2x)(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}} - \frac{2(\frac{\frac{-1}{2}(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}})cos(2x)}{sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2*-3cos(2x)*2cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{4}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2*-sin(2x)*2}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{2cos(2x)*-(\frac{-2cos(2x)*2cos^{2}(2x)}{sin^{3}(2x)} + \frac{-2cos(2x)sin(2x)*2}{sin^{2}(2x)} - \frac{2*-2cos(2x)*2cos(2x)}{sin^{3}(2x)} - \frac{2*-sin(2x)*2}{sin^{2}(2x)} + \frac{-2cos(2x)*2}{sin^{3}(2x)})*\frac{1}{2}}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{3}(2x)(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}}\\=&\frac{-4cos^{6}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{6}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{8cos^{4}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{4}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{16cos^{5}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{6}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{24cos^{3}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{4}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{24cos^{4}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{6}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{12cos^{4}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{4}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{16cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{2}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{4cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{2}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{8cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{2}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{24cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{4}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{16cos^{3}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{6}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{24cos^{3}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{4}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{20cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{2}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{8cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{4}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{4}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{4cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{6}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{12cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{4}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} - \frac{24cos^{4}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{6}(2x)} - \frac{4cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{2}(2x)} - \frac{4cos^{6}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{6}(2x)} - \frac{8cos^{4}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{4}(2x)} + \frac{16cos^{3}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{6}(2x)} - \frac{4}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{3}{2}}sin^{2}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{16cos^{5}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{6}(2x)} + \frac{8cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{2}(2x)} + \frac{24cos^{3}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{4}(2x)} - \frac{24cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{4}(2x)} + \frac{4}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{\frac{1}{2}}sin^{2}(2x)sqrt(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})} + \frac{8cos(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{4}(2x)} - \frac{4cos^{2}(2x)}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{6}(2x)} - \frac{4}{(\frac{cos^{2}(2x)}{sin^{2}(2x)} - \frac{2cos(2x)}{sin^{2}(2x)} + \frac{1}{sin^{2}(2x)})^{2}sin^{2}(2x)}\\ \end{split}\end{equation} \]