Terms in Fourier Series

Question

Can any one explain why?

$$\int_0^\pi \sin(nx)\sin(mx)\,dx=\begin{cases}0,&n\not=m,\\ {\pi\over 2},&n=m,\end{cases}$$

and $$\int_0^\pi \cos(nx)\cos(mx)\,dx=\begin{cases} 0, &n\not=m,\\ {\pi\over 2}, &n=m,\end{cases}$$

Answer 1

HINT:

Use Werner Formulas $$2\sin mx\sin nx=\cos(m-n)x-\cos(m+n)x$$

$$2\cos mx\cos nx=\cos(m-n)x+\cos(m+n)x$$

See the difference in behavior for $m=n$

Answer 2

Observe that the set $\{1,sinx,sin2x,...sinmx,cosx,...,cosmx\}$ is an orthonormal basis of the space of $2\pi$ -periodic functions ,let's say $T_m$,which is a Hilbert space with inner product $<f,g>=\frac {1}{\pi}\int_{-\pi}^{\pi} f\cdot g$. The only way i know to prove it ,is by using the formula above,though.