Prove each of the following`sin^(- 1)(3/5)+sin^(- 1)(8/17)=sin^(- 1)(77/85)`

Explanation

\begin{aligned}
&\sin ^{-1} \frac{3}{5}+\sin ^{-1} \frac{8}{17} \\ \\
&\sin ^{-1}\left\{\frac{3}{5} \sqrt{1-\left(\frac{8}{17}\right)^{2}}+\frac{8}{17} \sqrt{1-\left(\frac{3}{5}\right)^{2}}\right\} \\ \\
&\sin ^{-1}\left\{\frac{3}{5} \times \frac{15}{17}+\frac{8}{17} \times \frac{4}{5}\right\} \\ \\
&\sin ^{-1}\left\{\frac{45+32}{85}\right\}=\sin ^{-1} \frac{77}{85}
\end{aligned}