Write the value of ^ -1 [2 (2 ^ -1 3 2 ) ]. - Vidyadip

Question

Write the value of $\tan^{-1}\bigg[2\sin\bigg(2\cos^{-1}\frac{\sqrt{3}}{2}\bigg)\bigg].$

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Answer

$\tan^{-1}\bigg[2\sin\bigg(2\cos^{-1}\frac{\sqrt{3}}{2}\bigg)\bigg] = \tan^{-1}\bigg(2\sin\bigg(2\times\frac{\pi}{6}\bigg)\bigg)\bigg[\because\cos^{-1}\frac{\sqrt{3}}{2} = \frac{\pi}{6}\bigg]$
$ = \tan^{-1}\bigg(2\sin\frac{\pi}{3}\bigg) = \tan^{-1}\bigg(2\times\frac{\sqrt{3}}{2}\bigg) = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3}.$

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