Let be $\left<f,g\right>:=\frac{1}{\pi}\int\limits_{-\pi}^{\pi}f(x)g(x)dx$, where $f$ and $g$ are $2\pi$-periodic and Riemann integrable functions, and $\frac{1}{2}a_0+\sum\limits_{k=1}^na_k\cos(kx)+b_k\sin(kx)$ a partial sum of the Fourier series of $f$.
Then, we know that the Fourier coefficients $a_k$ of $f$ are defined by $$a_0:=\left<f,\cos(x\cdot 0)\right>\text{ and }a_k:= \left<f,\cos(x\cdot k)\right>.$$
After some tricky algebraic manipulations I have spotted that
$$\left<f-a_0,f-a_0\right> =\cdots=\left<f+a_0,f-a_0\right>,$$ which seems a bit odd. Is there any intuition behind this result? Or should I simply continue and not pay much attention to this detail?