Prove the following about the integral of Fourier coefficients

Question

I'm having a difficult time going from $$\sum_{n=1}^\infty (\cos nx \int_{-\pi}^\pi f(t) \cos nt dt + sin nx \int_{-\pi}^\pi f(t) \sin nt dt)$$ to $$\sum_{n=1}^{\infty}(\int_{-\pi}^\pi f(t) \cos n(t-x) dt)$$

Any help?

Answer 1

It's just $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$ and the fact that $\cos/\sin$ are even/odd respectively:

$$\cos(n(t-x)) = \cos(nt-nx)=\cos(nt)\cos(nx)+\sin(nt)\sin(nx)$$