Question

\[\int \left( \tan x + \cot x \right)^2 dx\]

Answer 1

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