Given,
$\sin ^{-1}\left(\frac{x}{13}\right)+\operatorname{cosec}^{-1}\left(\frac{13}{12}\right)=\frac{\pi}{2} ..(i)$
Let $\operatorname{cosec}^{-1} \frac{13}{12}=y$
Then. $\operatorname{cosec} y=\frac{13}{12} \Rightarrow \sin y=\frac{12}{13}$
$\therefore \cos y=\sqrt{1-\sin ^{2} y}$
$\begin{array}{l}
=\sqrt{1-\left(\frac{12}{13}\right)^{2}}=\sqrt{1-\frac{144}{169}} \\
=\sqrt{\frac{25}{169}}=\frac{5}{13} \Rightarrow y=\cos ^{-1} \frac{5}{13}
\end{array}$
Eq. (i) becomes, $\sin ^{-1}\left(\frac{x}{13}\right)+\cos ^{-1}\left(\frac{5}{13}\right)=\frac{\pi}{2}$
We know that
$\sin ^{-1} \theta+\cos ^{-1} \theta=\frac{\pi}{2}$
$\therefore$ Both angles of Eq. (i) should be same.
$\therefore x=5$