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

Formula

\(\sin^{-1}(\sin x)=(-1)^n(x-n\pi)\) for \(x\in\left[n\pi-\dfrac{\pi}{2},\ n\pi+\dfrac{\pi}{2}\right],\ n\in\mathbb{Z}\)

Variables: \(x\in\mathbb{R}\).

Conditions: Piecewise reduction to principal range of \(\sin^{-1}\).

Exact evaluation of inverse-trig compositions.