How to Integrate [ tan(x) + cot(x) ]^2 ?

[mauriciohideki]

\[\int_{}^{}(\tan x + \cot x)^{2}dx\]
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[Deleted member 4993]

\[\int_{}^{}(\tan x + \cot x)^{2}dx\]
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Do you know how to integrate:

\(\displaystyle \displaystyle{\int tan^2(\theta)d\theta}\)

 

[Prove It]

Hints: \(\displaystyle \displaystyle \begin{align*} \tan^2{(x)} + 1 \equiv \sec^2{(x)} \end{align*}\) and \(\displaystyle \displaystyle \begin{align*} 1 + \cot^2{(x)} \equiv \csc^2{(x)} \end{align*}\). Which of these trigonometric squares have easy antiderivatives.

It would also help you to expand the whole expression out.