$I_n=\int _0^{\frac{\pi }{4}}\tan ^n\left(x\right)dx,\: n \ge 2$
I have to find the monotony of $I_n$.
I calculated and proved $I_n + I_{n+2} = \frac{1}{n+1}$. Also, the first term $I_2 = 1 - \frac{\pi}{4}$
But I don't know how to find $I_{n+1}-I_{n}$ or if that's the way to go.
For every $x$ in the range, $\tan x<1$, so that $\tan^{n+1}x<\tan^n x$.