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I'm working my way through a book for prelim prep and found the problem: Find the fourier series for $\cos^{2N}(\theta )$. The hint is to not use integrals but the only method I know involves $\frac{1}{2\pi} \int_0^{2\pi} f(x)e^{-inx}dx$

Am I missing something about this expression that makes it solvable without the integral?

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Euler's identity will help you expand $\cos^{2N}\theta$ in terms of trigonometric polynomial. Then use the identity back to simplify them. For example, if $N = 1$, then

$$ \cos^{2}\theta = \left( \frac{e^{i\theta} + e^{-i\theta}}{2} \right)^{2} = \frac{e^{2i\theta} + 2 + e^{-2i\theta}}{4}= \frac{1}{2} + \frac{\cos2\theta}{2}, $$

which is the Fourier expansion of $\cos^{2}\theta$.

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The following might help you out when determining the integrals without complete computation.

Hint: Think $e^{-inx}$ of Euler's identity. We get $\cos(nx) - i\sin(nx)$. Now, look at the integral carefully. If we multiply each term by term and then rewrite the whole integral as two integrals, what do you know about each expression in the integral?

Hint: The integral of even function over interval with symmetric bounds is zero, whereas the integral of odd function over interval with symmetric bounds is not zero.

Hint: The product of two even functions is an even function. The product of two odd functions is an even function. Also, the product of an even function and an odd function is an odd function.

Hint: $\cos$ is an even function, whereas $\sin$ is an odd function.

See the question asked here in MSE site and reference about even and odd functions.

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