Prove that $\int_{0}^{\frac{\pi}{2}}{\frac{\sin(2n+1)x}{\sin(x)}dx}=\frac{\pi}{2}$ for $n\ge0$ Prove that
$$\int_{0}^{\frac{\pi}{2}}{\frac{\sin(2n+1)x}{\sin(x)}dx}=\frac{\pi}{2}$$ for $n\ge0$
I am not able to proceed with the integral. For the case $k+1$ please guide me through the problem. 
Thanks in advance!

Hint:
$$
\frac{1}{2} + \cos y + \cdots +\cos (ny) = \frac{\sin (n+1/2)y}{2 \sin (y/2)}.
$$
This is a useful identity that can be used to prove the convergence of Fourier series. It is proved by induction on $n \in \mathbb{N}$.
