If tan \left ( cos^{-1}x \right )= Sin\left ( cot^{-1}\frac{1}{2} \right )   then x is equal to

  • Option 1)

    \pm \frac{5}{3}

  • Option 2)

    \pm \frac{\sqrt{5}}{3}

  • Option 3)

    \pm \frac{5}{\sqrt{3}}

  • Option 4)

    None of these

 

Answers (1)

As we learnt in 

Relation between all the Inverse Trigonometric Functions -

\sin ^{-1}x = \tan ^{-1}\frac{x}{\sqrt{1-x^{2}}} = cosec ^{-1}\frac{1}{x} = \cos ^{-1}\sqrt{1-x^{2}} = \cot ^{-1}\frac{\sqrt{1-x^{2}}}{x} = \sec ^{-1}\frac{1}{\sqrt{1-x^{2}}}

- wherein

This happens with domain of functions, x>0

 

 tan (cos^{-1}x)\:=\:sin(cot^{-1}\frac{1}{2})

\frac{\sqrt{1-x^{2}}}{x}\:=\:\frac{2}{\sqrt{5}}

\therefore 4x^{2}=5-5x^{2}

\Rightarrow x^{2}=\frac{5}{3} \Rightarrow x=\:\frac{\pm \sqrt{5}}{3}

 


Option 1)

\pm \frac{5}{3}

Incorrect

Option 2)

\pm \frac{\sqrt{5}}{3}

Correct

Option 3)

\pm \frac{5}{\sqrt{3}}

Incorrect

Option 4)

None of these

Incorrect

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