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. $4$
• B. $5$
• C. $1$
• D. $3$

Answer 1

The correct option is D $3$

${\sin }^{-1}\frac{x}{5}+{\text{cosec}}^{-1}\frac{5}{4}=\frac{\pi }{2}\Rightarrow {\text{sin}}^{-1}\frac{x}{5}=\frac{\pi }{2}-{\text{cosec}}^{-1}\frac{5}{4}$
$\Rightarrow {\text{sin}}^{-1}\frac{x}{5}=\frac{\pi }{2}-{\text{sin}}^{-1}\frac{4}{5}$
$\Rightarrow {\text{sin}}^{-1}\frac{x}{5}={\text{cos}}^{-1}\frac{4}{5}[\because {\sin }^{-1}y+{\cos }^{-1}y=\frac{\pi }{2}]$
$\Rightarrow {\text{sin}}^{-1}\left(\frac{x}{5}\right)={\text{sin}}^{-1}\frac{3}{5}$
$\therefore x=3$