Evaluate the following: (2 ^ -1 2 3 )+ ( ^ -1 3 ) - Vidyadip

Question

Evaluate the following:
$\sin\Big(2\tan^{-1}\frac{2}{3}\Big)+\cos\Big(\tan^{-1}\sqrt3\Big)$

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Answer

$\sin\Big(2\tan^{-1}\frac{2}{3}\Big)+\cos\Big(\tan^{-1}\sqrt3\Big)$
$=\sin\Bigg(\sin^{-1}\frac{2\times\frac{2}{3}}{1+\frac{4}{9}}\Bigg)+\cos\Bigg(\cos^{-1}\frac{1}{\sqrt{1+\big(\sqrt3\big)^2}}\Bigg)$
$=\sin\Big(\sin^{-1}\frac{12}{13}\Big)+\cos\Big(\cos^{-1}\frac{1}{2}\Big)$
$=\sin\Big(\sin^{-1}\frac{12}{13}\Big)+\cos\Big(\cos^{-1}\frac{1}{2}\Big)$
$=\frac{12}{13}+\frac{1}{2}$
$=\frac{37}{26}$

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