# Fourier series of $\cos^4(x)$

Expand $$\cos^4(x)$$ into a Fourier series.

we already know that we need to find $$\int_{-\pi}^\pi f(x)dx$$ which will = $$\frac{3\pi}{4}$$.

now we need to find $$a_n = \int_{-\pi}^\pi \cos^4(x) \cos(nx) dx$$ and $$b_n = \int_{-\pi}^\pi cos^4(x) \sin(nx) dx$$.

and we already know $$\cos^4$$ is an even function so $$b_n = 0$$. So how to find $$a_n$$.

Thanks.

$$\cos^2(x)=\frac{1+\cos(2x)}{2}.$$