Chapter 5 - Inner Product Spaces - 5.2 Inner Product Spaces - 5.2 Exercises - Page 246: 68

see the proof below.

Let $f(x)= 1, g(x)=\cos 2nx) $, $C[0,\pi]$ \begin{aligned}\langle f,g\rangle &=\int_{0}^{\pi} \cos 2nx d x\\ &=\left[\frac{1}{2n} \cos 2nx\right]_{0}^{\pi} \\ &=0 \end{aligned}. then $f$ and $g$ are orthogonal in the inner product space $C[0,\pi]$.