Project $f(x) = \sin^2(x)$ onto the plane spanned by $\sin(2x)$ and $\cos(2x)$

Question

I approached the problem by projecting $f(x)$ onto each of the other functions (since they are an orthonormal basis) using the formula $$\langle f(x),\sin2x\rangle \sin2x + \langle f(x),\cos2x\rangle \cos2x.$$

Before I go through the computations, I wanted to make sure this is the correct approach. Thanks!

Answer 1

If you're working with the Fourier series (which is likely, because the two basis you give are orthogonal Fourier bases), your formula is how the projection should work.

Observation: If you're not required to show the process, just notice that $$ \sin^2x = 1-\cos^2x = 1-\frac{1+\cos2x}{2} = 1-\frac{1+\cos2x}{2} = \frac{1}{2} - \frac{1}{2}\cos 2x. $$ The extra constant $\displaystyle\frac{1}{2}$ is for the basis of $1$, which is orthogonal to both $\sin 2x$ and $\cos 2x$.