$\text { L.H.S. } =\sin ^{-1}\left(\frac{3}{5}\right)-\sin ^{-1}\left(\frac{8}{17}\right)$
$=\cos ^{-1} \sqrt{1-\frac{9}{25}}-\cos ^{-1} \sqrt{1-\frac{64}{289}}$
$=\cos ^{-1}\left(\frac{4}{5}\right)-\cos ^{-1}\left(\frac{15}{17}\right)$
$=\cos ^{-1}\left[\frac{4}{5} \times \frac{15}{17}+\sqrt{1-\frac{16}{25}} \sqrt{1-\frac{225}{289}}\right]$
$=\cos ^{-1}\left[\frac{60}{85}+\frac{3}{5} \times \frac{8}{17}\right]$
$=\cos ^{-1}\left[\frac{60}{85}+\frac{24}{85}\right]$
$=\cos ^{-1}\left(\frac{84}{85}\right)=\text { R.H.S. }$
