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  4. If cos-1 ((2/3x))+ cos-1 ((3/4x)) = (π/2) (x > (3/4)) then x is equal to :

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Q. If $\cos^{-1} \left(\frac{2}{3x}\right)+\cos^{-1} \left(\frac{3}{4x}\right) = \frac{\pi}{2} \left(x > \frac{3}{4}\right) $ then $x$ is equal to :

JEE MainJEE Main 2019Inverse Trigonometric Functions

Solution:

$\cos^{-1} \left(\frac{2}{3x}\right) + \cos^{-1} \left(\frac{3}{4x}\right) = \frac{\pi}{2} \left(x> \frac{3}{4}\right) $
$ \cos^{-1} \left(\frac{3}{4x}\right) = \frac{\pi}{2} -\cos^{-1} \left(\frac{2}{3x}\right) $
$ \cos^{-1} \left(\frac{3}{4x}\right)=\sin^{-1} \left(\frac{2}{3x}\right) $
$\cos\left(\cos^{-1} \left(\frac{3}{4x}\right)\right) = \cos\left(\sin^{-1} \frac{2}{3x}\right)$
$ \frac{3}{4x} = \frac{\sqrt{9x^{2}-4}}{3x}$
$ \frac{81}{16} +4 = 9x^{2}$
$x^{2} = \frac{145}{16\times9}\Rightarrow x = \frac{\sqrt{145}}{12} $