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  4. tan 2( sec -1 3)+ textcosec2( cot -1 2)+ cos 2( cos -1 (2/3)+ sin -1 (2/3))= .

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Q. $\tan ^{2}\left(\sec ^{-1} 3\right)+\text{cosec}^{2}\left(\cot ^{-1} 2\right)+\cos ^{2}\left(\cos ^{-1} \frac{2}{3}+\sin ^{-1} \frac{2}{3}\right)=$ _________.

Gujarat CETGujarat CET 2020

Solution:

$\tan ^{2}\left(\sec ^{-1} 3\right)+\text{cosec}^{2}\left(\cot ^{-1} 2\right)+\cos ^{2}\left(\cos ^{-1} \frac{2}{3}+\sin ^{-1} \frac{2}{3}\right)$
$=\tan ^{2}\left(\tan ^{-1}(2 \sqrt{2})\right)+\text{cosec}^{2}\left(\text{cosec}^{-1} \sqrt{5}\right)+\cos ^{2}\left(\frac{\pi}{2}\right)$
$=8+5=13$