[Math] use parseval’s identity to evaluate the integral $ \int_{-\pi}^{\pi}\sin^4 xdx$

Question

use parseval's identity to evaluate the integral

\begin{equation} \int_{-\pi}^{\pi}(\sin x)^4dx\end{equation}

I'm familiar with Parseval's identity which states that for each piecewise continuous complex function $f$ we have the equality
\begin{equation}
\int_{-\pi}^{\pi}\left|f(x) \right|^{2}dx=\frac{|a_{0}|^{2}}{2}+\sum_{n=1}^{\infty}\left(|a_{n}|^{2}+|b_{n}|^{2} \right)
\end{equation}
where $a_{n}$ and $b_{n}$ are the Fourier coefficients of $f$.

I'm confused how evaluate $\sin^4x$

Answer 1

We have: $$\sin^2(x) = \frac{1-\cos(2x)}{2}\tag{1}$$ hence Parseval's identity implies: $$ \int_{-\pi}^{\pi}\sin^4(x)\,dx = 2\pi\cdot\frac{1}{4}+\pi\cdot\frac{1}{4}=\color{red}{\frac{3\pi}{4}}.\tag{2}$$