Why $\int_{0}^{\pi}\arctan{\cos{x}}dx = 0$?

Question

I saw this in Ron Gordon's answer to this question:

I need assistance in integrating $ \frac{x \sin x}{1+(\cos x)^2}$

Thank you!

Answer 1

Odd around $\pi/2.$ That is, given $f(x) = \arctan \cos x,$ we have $f(\pi - x) = -f(x).$ Draw a graph.