Question

If $tan\left(se{c}^{-1}x\right)=sin\left(co{s}^{-1}\frac{1}{\sqrt{5}}\right)$, then x is equal to

• A. $\pm \frac{3}{\sqrt{5}}$
• B. $\pm \frac{\sqrt{5}}{3}$
• C. $\pm \sqrt{\frac{3}{5}}$
• D. None of these

Answer 1

The correct option is A $\pm \frac{3}{\sqrt{5}}$

$tan\left(se{c}^{-1}x\right)=sin\left(co{s}^{-1}\frac{1}{\sqrt{5}}\right)$
$\Rightarrow \sqrt{se{c}^2\left(se{c}^{-1}x\right)-1}=\sqrt{1-co{s}^2\left(co{s}^{-1}\frac{1}{\sqrt{5}}\right)}$
$\Rightarrow \sqrt{x^2-1}=\sqrt{1-\frac{1}{5}}$
$\Rightarrow x^2-1=\frac{4}{5}$
$\Rightarrow x^2=\frac{9}{5}$
$\Rightarrow x=\pm \frac{3}{\sqrt{5}}$