Mathematics - Inverse Trigonometric Functions Question with Solution | TestHub

Consider ${\tan }^{-1}\left[\sin \left({\cos }^{-1}\sqrt{\frac{2}{3}}\right)\right]$ $$\begin{array}{rl}& \text{ Let }{\cos }^{-1}\sqrt{\frac{2}{3}}=\theta \Rightarrow \cos \theta =\sqrt{\frac{2}{3}}\\ & \Rightarrow \sin \theta =\sqrt{1-{\cos }^2\theta }=\sqrt{1-\frac{2}{3}}=\sqrt{\frac{1}{3}}\\ & \therefore {\tan }^{-1}\left[\sin \left({\cos }^{-1}\sqrt{\frac{2}{3}}\right)\right]={\tan }^{-1}[\sin \theta ]\\ & ={\tan }^{-1}\left[\sqrt{\frac{1}{3}}\right]={\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)\\ & =\frac{\pi }{6}\end{array}$$