$\text { L.H.S. }=\cos ^{-1}\left(\frac{3}{5}\right)+\cos ^{-1}\left(\frac{12}{13}\right)+\cos ^{-1}\left(\frac{63}{65}\right)$
$=\cos ^{-1}\left[\frac{3}{5} \times \frac{12}{13}-\sqrt{1-\frac{9}{25}} \sqrt{1-\frac{144}{169}}\right]+\cos ^{-1}\left(\frac{63}{65}\right)$
$=\cos ^{-1}\left[\frac{36}{65}-\sqrt{\frac{16}{25}} \sqrt{\frac{25}{169}}\right]+\cos ^{-1}\left(\frac{63}{65}\right)$
$=\cos ^{-1}\left[\frac{36}{65}-\frac{4}{5} \times \frac{5}{13}\right]+\cos ^{-1}\left(\frac{63}{65}\right)$
$=\cos ^{-1}\left[\frac{36-20}{65}\right]+\cos ^{-1}\left(\frac{63}{65}\right)$
$=\cos ^{-1}\left(\frac{16}{65}\right)+\cos ^{-1}\left(\frac{63}{65}\right)$
$=\cos ^{-1}\left[\frac{16}{65} \times \frac{63}{65}-\sqrt{1-\left(\frac{16}{65}\right)^2} \sqrt{1-\left(\frac{63}{65}\right)^2}\right]$
$=\cos ^{-1}\left[\frac{1008}{(65)^2}-\sqrt{1-\frac{256}{4225}} \sqrt{1-\frac{3969}{4225}}\right]$
$=\cos ^{-1}\left[\frac{1008}{(65)^2}-\sqrt{\frac{4225-256}{4225}} \sqrt{\frac{4225-3969}{4225}}\right]$
$=\cos ^{-1}\left[\frac{1008}{(65)^2}-\sqrt{\frac{3969}{4225}} \sqrt{\frac{256}{4225}}\right]$
$=\cos^{-1}[\frac {1008}{(65)^2}-\frac {63}{65}×\frac {16}{65}]$
$=\cos ^{-1}\left[\frac{1008}{(65)^2}-\frac{1008}{(65)^2}\right]=\cos ^{-1}(0)=\frac{\pi}{2}=\text { R.H.S. }$
