Question

If ${\sin }^{-1}\left(\frac{x}{5}\right)+{\text{cosec}}^{-1}\left(\frac{5}{4}\right)=\frac{\pi }{2}$ , then the value of x is-

• A. $1$
• B. $3$
• C. $4$
• D. $5$

Answer 1

The correct option is B $3$

${\sin }^{-1}\left(\frac{x}{5}\right)+{\text{cosec}}^{-1}\left(\frac{5}{4}\right)=\frac{\pi }{2}$
$\Rightarrow {\sin }^{-1}\left(\frac{x}{5}\right)+{\sin }^{-1}\left(\frac{4}{5}\right)=\frac{\pi }{2}....\left(i\right)$
As we know,
${\sin }^{-1}\left(\frac{4}{5}\right)=x\Rightarrow \sin x=\frac{4}{5}\Rightarrow 1-{\sin }^{2}x=1-\frac{16}{25}\Rightarrow {\cos }^{2}x=\frac{9}{25}\Rightarrow \cos x=\frac{3}{5}\Rightarrow x={\cos }^{-1}\frac{3}{5}$
Substituting in $\left(i\right)$
$\Rightarrow {\sin }^{-1}\left(\frac{x}{5}\right)+{\cos }^{-1}\left(\frac{3}{5}\right)=\frac{\pi }{2}$
$\because {\sin }^{-1}\theta +{\cos }^{-1}\theta =\frac{\pi }{2}$
$\therefore \frac{x}{5}=\frac{3}{5}\Rightarrow x=3$