If si n ^ - 1 ((5)/(x)) + si n ^ - 1 ((12)/(x)) = si n ^ - 1...

si{n}^{- 1}\left(\dfrac{5}{x}\right)$and$si{n}^{- 1}\left(\dfrac{12}{x}\right)$are both defined when$x > 12 .$Therefore<br />$co{s}^{- 1}\left(\sqrt{1 - \dfrac{25}{{x}^{2}}}\right) + si{n}^{- 1}\left(\dfrac{12}{x}\right) = \dfrac{\pi}{2}$\left\{∵ si{n}^{- 1}\dfrac{2}{x} + co{s}^{- 1}\dfrac{2}{x} = \dfrac{\pi}{2}\right\}$
$si{n}^{- 1}\left(\dfrac{12}{x}\right) = si{n}^{- 1}\sqrt{1 - \dfrac{25}{{x}^{2}}}$
On comparing, we get,
$\dfrac{12}{x} = \sqrt{1 - \dfrac{25}{{x}^{2}}}$

Therefore, $144 = {x}^{2} - 25$ or $x = 13$