Mathematics - Inverse Trigonometric Functions Question with Solution | TestHub

Set $\theta ={\cos }^{-1}\mathrm{x}\in (0,\pi /2)$. Then $\tan (\theta /2)=\sin \theta /(1+\cos \theta )=\sqrt{\left(1-x^2\right)}/(1+x)$, so the stated arctangent is $\theta /2$. The positive half-angle identities give both ${\cos }^{-1}(\sqrt{((1+x)/2)})$ and ${\sin }^{-1}(\sqrt{((1-x)/2)})$.