If $ tan(sec^{-1}x)= sin (cos^{-1}\frac{

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Target Exam

CUET

Subject

-- Mathematics - Section B1

Chapter

Inverse Trigonometric Functions

Question:

If $tan(sec^{-1}x)= sin (cos^{-1}\frac{1}{\sqrt{5}}),$ then x =

Options:

$±\frac{3}{\sqrt{5}}$

$±\frac{\sqrt{5}}{3}$

$±\sqrt{\frac{3}{5}}$

none of these

Correct Answer:

$±\frac{3}{\sqrt{5}}$

Explanation:

We have,

$tan(sec^{-1}x)= sin (cos^{-1}\frac{1}{\sqrt{5}})$

$⇒ \sqrt{sec^2(sec^{-1}x)-1}= \sqrt{1-cos^2 (cos^{-1}\frac{1}{\sqrt{5}})}$

$⇒ \sqrt{x^2-1} =\sqrt{1-\frac{1}{5}}⇒ x^2 -1 =\frac{4}{5} ⇒ x = ±\frac{3}{\sqrt{5}}$