Question

Evaluate: $\cos \left({\sin }^{-1}\frac{3}{5}+{\sin }^{-1}\frac{5}{13}\right)$

Answer 1

$$\begin{gathered} \cos \left({\sin }^{-1}\frac{3}{5}+{\sin }^{-1}\frac{5}{13}\right)=\cos \left\{{\sin }^{-1}\left(\frac{3}{5}\sqrt{1-{\left(\frac{5}{13}\right)}^2}+\frac{5}{13}\sqrt{1-{\left(\frac{3}{5}\right)}^2}\right)\right\}\left[\because {\sin }^{-1}x+{\sin }^{-1}y={\sin }^{-1}\left(x\sqrt{1-y^2}+y\sqrt{1-x^2}\right)\right] \\ =\cos \left\{{\sin }^{-1}\left(\frac{3}{5}\times \frac{12}{13}+\frac{5}{13}\times \frac{4}{5}\right)\right\} \\ =\cos \left\{{\sin }^{-1}\left(\frac{36}{65}+\frac{4}{13}\right)\right\} \\ =\cos \left\{{\sin }^{-1}\left(\frac{56}{65}\right)\right\} \\ =\cos \left\{{\cos }^{-1}\sqrt{1-{\left(\frac{56}{65}\right)}^2}\right\}\left[\because {\sin }^{-1}x={\cos }^{-1}\sqrt{1-x^2}\right] \\ =\cos \left\{{\cos }^{-1}\frac{33}{65}\right\} \\ =\frac{33}{65} \end{gathered}$$