Question

$\int {\left(\tan x+\cot x\right)}^{2}dx$

Answer 1

$$\begin{gathered} \int {\left(\tan x+\cot x\right)}^2 \\ =\int \left({\tan }^{2}x+{\cot }^{2}x+2\tan x\cot x\right)dx \\ =\int \left({\tan }^{2}x+{\cot }^{2}x+2\right)dx \\ =\int \left[\left({\sec }^2\mathrm{x}-1\right)+\left({\mathrm{cosec}}^{2}x-1\right)+2\right]dx \\ =\int \left({\sec }^{2}x+{\mathrm{cosec}}^{2}x\right)dx \\ =\tan x-\cot x+\mathrm{C} \end{gathered}$$