Question

If ${\sin }^{-1}\left(\frac{x}{13}\right)+co{\sec }^{-1}\left(\frac{13}{12}\right)=\frac{\pi }{2}$, then the value if $x$ is

• A. $5$
• B. $4$
• C. $12$
• D. $11$

Answer 1

The correct option is A $5$

${\sin }^{-1}\left(\frac{x}{13}\right)+{\csc }^{-1}\left(\frac{13}{12}\right)=\frac{\pi }{2}$

$\Rightarrow$${\sin }^{-1}\left(\frac{x}{13}\right)-\frac{\pi }{2}=-{\csc }^{-1}\left(\frac{13}{12}\right)$

$\Rightarrow$${\sin }^{-1}\left(\frac{x}{13}\right)-{\sin }^{-1}\left(1\right)=-{\sin }^{-1}\left(\frac{12}{13}\right)$

$\Rightarrow {\sin }^{-1}(\frac{x}{13}\sqrt{1-1}-1\sqrt{1-\frac{x^2}{169}})={\sin }^{-1}(-\frac{12}{13})$

$\Rightarrow (-1\sqrt{1-\frac{x^2}{169}})=(-\frac{12}{13})$

Squaring both sides, we get

$\Rightarrow 1-\frac{x^2}{169}=\frac{144}{169}$

$\Rightarrow x^2=169-144$

$\Rightarrow x=\pm 5$

Hence, A is the correct option.