Question

If $sin^{-1}\left(\frac{5}{x}\right) + sin^{-1}\left(\frac{12}{x}\right) = \frac{\pi}{2}$, then $x$ is equal to _____.

Answer 1

Put $sin^{-1} \frac{5}{x} = A$ or $\frac{5}{x} = sin\,A$
$sin^{-1} \frac{12}{x} = B$ or $\frac{12}{x} = sin\,B$
$\Rightarrow A + B = \frac{\pi}{2}$
$sin\,A = sin(\frac{\pi}{2} - B) = cos\,B = \sqrt{1 - sin^2\,B}$
$\Rightarrow \frac{5}{x} = \sqrt{1 - \frac{144}{x^2}}$ or $\frac{169}{x^2} = 1$
or $x^2 = 169$ or $x = 13$
[$\because x = -13$ does not satisfy the given equation]