How to prove $$\frac{1}{2n+2}<\int\_0^{\frac{\pi}{4}}\tan^nx\,dx< \frac{1}{2n}$$
Set $A\_n=\int\_0^{\frac{\pi}{4}}\tan^nx\,dx$, then we have $A\_n+A\_{n+2}=\frac{1}{n+1}$ and we have $A\_{n+2} < A\_n$ ,so we can get $$\frac{1}{2n+2}< \int\_0^{\frac{\pi}{4}}\tan^nx\,dx < \frac{1}{2n-2}$$
But how to show that$$\int\_0^{\frac{\pi}{4}}\tan^nx\,dx < \frac{1}{2n}$$