Inverse trigonometric principal domains and ranges for JEE

Formula

\[\begin{aligned}&y=\sin^{-1}x\iff x=\sin y,\ y\in\left[-\frac\pi2,\frac\pi2\right],\ x\in[-1,1]\\&y=\cos^{-1}x\iff x=\cos y,\ y\in[0,\pi],\ x\in[-1,1]\\&y=\tan^{-1}x\iff x=\tan y,\ y\in\left(-\frac\pi2,\frac\pi2\right),\ x\in\mathbb R\\&y=\cot^{-1}x\iff x=\cot y,\ y\in(0,\pi),\ x\in\mathbb R\\&y=\sec^{-1}x\iff x=\sec y,\ y\in[0,\pi]\setminus\left\{\frac\pi2\right\},\ |x|\ge 1\\&y=\csc^{-1}x\iff x=\csc y,\ y\in\left[-\frac\pi2,\frac\pi2\right]\setminus\{0\},\ |x|\ge 1\end{aligned}\]

Variables: Principal values are chosen to make each inverse single-valued.

Conditions: Use principal branch conventions.

Inverse trig equations, graph, differentiation and continuity questions.