Solution: Let $z=\cos ^{-1} \frac{12}{13}+\sin ^{-1} \frac{4}{5}-\tan ^{-1} \frac{63}{16}$
$\because \quad \sin ^{-1} \frac{4}{5}=\frac{\pi}{2}-\cos ^{-1} \frac{4}{5} \\$
$\therefore \quad z=\cos ^{-1} \frac{12}{13}+\left(\frac{\pi}{2}-\cos ^{-1} \frac{4}{5}\right)-\tan ^{-1} \frac{63}{16} . \\$
$\quad z=\frac{\pi}{2}-\left(\cos ^{-1} \frac{4}{5}-\cos ^{-1} \frac{12}{13}\right)-\tan ^{-1} \frac{63}{16} \\$
$\quad \frac{4}{5}>0, \frac{12}{13}>0 \text { and } \frac{4}{5}<\frac{12}{13} \\$
$\therefore \quad \cos ^{-1} \frac{4}{5}-\cos ^{-1} \frac{12}{13}=\cos ^{-1}\left[\frac{4}{5} \times \frac{12}{13}+\sqrt{1-\frac{16}{25}} \cdot \sqrt{1-\frac{144}{169}}\right]=\cos ^{-1}\left(\frac{63}{65}\right)$
$\therefore \quad$ equation (i) can be written as
$z=\frac{\pi}{2}-\cos ^{-1}\left(\frac{63}{65}\right)-\tan ^{-1}\left(\frac{63}{16}\right)$
$z=\sin ^{-1}\left(\frac{63}{65}\right)-\tan ^{-1}\left(\frac{63}{16}\right)$
$\because \quad \sin ^{-1}\left(\frac{63}{65}\right)=\tan ^{-1}\left(\frac{63}{16}\right)$
$\therefore \quad$ from equation (ii), we get
$\therefore \quad z=\tan ^{-1}\left(\frac{63}{16}\right)-\tan ^{-1}\left(\frac{63}{16}\right) \quad \Rightarrow \quad z=0$
