Given Fourier series for $f(x)$ continuous over $[-\pi, \pi]$. find $\int_{-\pi}^\pi f(x)\cos^2(nx)dx$

Question

I'm learning to the exam and I find this exercise in the book , and I can't think how I solve it.

Given Fourier series for $f(x)$ continuous over $[-\pi, \pi]$.

$$f(x) \approx \frac{a_0}{2} + \sum^\infty_{n=1}a_n\cos(nx) + b_n\sin(nx).$$

Find $\int_{-\pi}^\pi f(x)\cos^2(nx)dx$.

Can you please give me some hints to solve it? thanks in advance.

Answer 1

As mentioned in comments write $\displaystyle \cos^2(nx) = \frac{\cos (2nx) + 1}{2}$. Use the series value of right hand side and use orthogonality of sines and cosines in $[-\pi, \pi]$. You should get $a_{2n}/2$.