Q. If $\sin^{-1} \left(\frac{x}{5}\right) + cosec^{-1} \left(\frac{5}{4}\right) = \frac{\pi}{2} $, then the values of x is
Inverse Trigonometric Functions
Solution:
$\sin^{-1} \left(\frac{x}{5}\right) + cosec^{-1} \left(\frac{5}{4}\right) = \frac{\pi}{2}$
$ \Rightarrow \sin^{-1} \left(\frac{x}{5}\right) = \frac{\pi}{2} -cosec^{-1} \left(\frac{5}{4}\right) $
$ \Rightarrow \sin^{-1}\left(\frac{x}{5} \right) = \frac{\pi}{2} - \sin^{-1} \left(\frac{4}{5}\right) $
$ \left[\because \sin^{-1} x + \cos^{-1}x = \pi/2\right] $
$ \Rightarrow \sin^{-1}\left(\frac{x}{5}\right) = \cos^{-1} \left(\frac{4}{5}\right)$ ...(1)
Let $ \cos^{-1} \frac{4}{5} = A \Rightarrow \cos A = \frac{4}{5} $
$ \Rightarrow A = \cos^{-1} \left(4/5\right) $
$ \Rightarrow \sin A = \frac{3}{5}$
$ \Rightarrow \sin^{-1} \frac{3}{5}$
$ \therefore \cos^{-1} \left(4/5\right) =\sin^{-1}\left(3/5\right) $
$ \therefore $ equation (1) become,
$ \sin^{-1} \frac{x}{5} = \sin^{-1} \frac{3}{5} \Rightarrow \frac{x}{5} = \frac{ 3 }{5} \Rightarrow x= 3 $
