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When deriving Fourier series, an important step is to establish that the integral of the product of two periodic sin functions is 0 if they have a different frequency.

This then allows you to define Fourier series as the basis for an inner product space.

The usual derivation makes use of trig identities. Here's an example: Orthogonality of sine and cosine integrals.

However I don't find these derivations very intuitive.

Is there another way to see why the integral of sin/cos functions of different frequencies is always 0?

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