General principal value reduction for \(\cos^{-1}(\cos x)\) for JEE

Formula

\(\cos^{-1}(\cos x)=\begin{cases}x-2n\pi,&x\in[2n\pi,(2n+1)\pi]\\ 2(n+1)\pi-x,&x\in[(2n+1)\pi,2(n+1)\pi]\end{cases}\)

Variables: \(n\in\mathbb{Z}\).

Conditions: Output always in \([0,\pi]\).

Exact evaluation of compositions with cosine inverse.