If ^ - 1 x 5 + cose c ^ - 1 ( 5 4 ) = 2 , then x = - Vidyadip

MCQ

If ${\sin ^{ - 1}}\frac{x}{5} + {\rm{cose}}{{\rm{c}}^{ - 1}}\left( {\frac{5}{4}} \right) = \frac{\pi }{2},$ then $x = $

A
$4$
B
$5$
C
$1$
✓
$3$

✓

Answer

Correct option: D.

$3$

d
Given $\sin ^{-1}\left(\frac{x}{5}\right)+\csc ^{-1}\left(\frac{5}{4}\right)=\frac{\pi}{2}$

$\Rightarrow \sin ^{-1}\left(\frac{x}{5}\right)+\sin ^{-1}\left(\frac{4}{5}\right)=\frac{\pi}{2}$

$\Rightarrow \sin ^{-1}\left(\frac{x}{5}\right)=\frac{\pi}{2}-\sin ^{-1}\left(\frac{4}{5}\right)$

$\Rightarrow \sin ^{-1}\left(\frac{x}{5}\right)=\cos ^{-1}\left(\frac{4}{5}\right)$

$\Rightarrow \frac{x}{5}=\sin \left(\cos ^{-1} \frac{4}{5}\right)$

$\Rightarrow \frac{x}{5}=\sin \left(\sin ^{-1} \frac{3}{5}\right)$

$\Rightarrow \frac{x}{5}=\frac{3}{5}$

$\Rightarrow x=3$

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