1.

Find the value of \(2{\tan ^{ - 1}}\left[ {2\cos \left( {2{{\sin }^{ - 1}}\frac{{\sqrt 3 }}{2}} \right)} \right]\) .1. \( -\frac{\pi }{4}\)2. \( - \frac{\pi }{3}\)3. \(\frac{\pi }{4}\)4. \( - \frac{\pi}{2}\)

Answer» Correct Answer - Option 4 : \( - \frac{\pi}{2}\)

Concept:

If x = sin θ then \(θ = {\sin ^{ - 1}}x\)

Similarly, if x = tan θ then \(θ = {\tan ^{ - 1}}x\)

Calculation: 

The expression \({\tan ^{ - 1}}\left[ {2\cos \left( {2{{\sin }^{ - 1}}\frac{{\sqrt 3 }}{2}} \right)} \right]\) can be rewritten as:

\({\tan ^{ - 1}}\left[ {2\cos \left( {2{{\sin }^{ - 1}}\frac{{\sqrt 3 }}{2}} \right)} \right] = {\tan ^{ - 1}}\left[ {2\cos \left( {\frac{{2\pi }}{3}} \right)} \right]\)

\(= {\tan ^{ - 1}}\left[ {2\left( { - \frac{1}{2}} \right)} \right]\)

\(= {\tan ^{ - 1}}\left( { - 1} \right)\)

\(= - \frac{\pi}{4}\)

\(2{\tan ^{ - 1}}\left[ {2\cos \left( {2{{\sin }^{ - 1}}\frac{{\sqrt 3 }}{2}} \right)} \right]\) \(= - 2 \times \frac{\pi}{4} = - \frac{\pi}{2}\)


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