1. **Problem statement:** Find the exact value of each expression using a calculator or exact trigonometric identities. 2. **Part (a):** Calculate $\cos(\sin^{-1}(\frac{3}{5}))$. - Let $\theta = \sin^{-1}(\frac{3}{5})$, so $\sin \theta = \frac{3}{5}$. - Using the Pythagorean identity: $\cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5}$. - Since $\theta$ is in the range $[-\frac{\pi}{2}, \frac{\pi}{2}]$ for $\sin^{-1}$, $\cos \theta$ is positive. 3. **Part (b):** Calculate $\cos^{-1}\left(\sqrt{3} \tan \frac{\pi}{6}\right)$. - First, find $\tan \frac{\pi}{6} = \frac{1}{\sqrt{3}}$. - Then, $\sqrt{3} \times \frac{1}{\sqrt{3}} = 1$. - So, $\cos^{-1}(1) = 0$ because $\cos 0 = 1$. 4. **Part (c):** Calculate $\cot(\sin^{-1}(\frac{2}{3}))$. - Let $\phi = \sin^{-1}(\frac{2}{3})$, so $\sin \phi = \frac{2}{3}$. - Using Pythagorean identity: $\cos \phi = \sqrt{1 - \sin^2 \phi} = \sqrt{1 - \left(\frac{2}{3}\right)^2} = \sqrt{1 - \frac{4}{9}} = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3}$. - $\cot \phi = \frac{\cos \phi}{\sin \phi} = \frac{\frac{\sqrt{5}}{3}}{\frac{2}{3}} = \frac{\sqrt{5}}{3} \times \frac{3}{2} = \frac{\sqrt{5}}{2}$. **Final answers:** - (a) $\frac{4}{5}$ - (b) $0$ - (c) $\frac{\sqrt{5}}{2}$