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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.

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Big Hint

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

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