SOLUTION : Consider, R.H.S. $=\sin ^{-1}\left(\frac{5}{13}\right)+\cos ^{-1}\left(\frac{3}{5}\right)$
$=\sin ^{-1}\left(\frac{5}{13}\right)+\sin ^{-1}\left(\frac{4}{5}\right) {\left[\because \cos ^{-1} x=\sin ^{-1} \sqrt{1-x^2}\right]}$ \\
$=\sin ^{-1}\left[\frac{5}{13} \sqrt{1-\frac{16}{25}}+\frac{4}{5} \sqrt{1-\frac{25}{169}}\right] {\left[\sin ^{-1} x+\sin ^{-1} y=\sin -1\left[x \sqrt{1-y^2}+y \sqrt{1-x^2}\right]\right]} \\$
$=\sin ^{-1}\left[\frac{5}{13} \times \frac{3}{5}+\frac{4}{5} \times \frac{12}{13}\right]=\sin ^{-1}\left(\frac{63}{65}\right)=\text { L.H.S. } $
