Question

$50\tan \left(3{\tan }^{-1}\left(\frac{1}{2}\right)+2{\cos }^{-1}\left(\frac{1}{\sqrt{5}}\right)\right)+4\sqrt{2}\tan \left(\frac{1}{2}{\tan }^{-1}(2\sqrt{2})\right)$ is equal to _______.

Answer 1

Answer: 29

$50\tan \left(3{\tan }^{-1}\frac{1}{2}+2{\cos }^{-1}\frac{1}{\sqrt{5}}\right)$
$+4\sqrt{2}\tan \left(\frac{1}{2}{\tan }^{-1}2\sqrt{2}\right)$
$=50\tan \left({\tan }^{-1}\frac{1}{2}+2\left({\tan }^{-1}\frac{1}{2}+{\tan }^{-1}2\right)\right)$
$+4\sqrt{2}\tan \left(\frac{1}{2}{\tan }^{-1}2\sqrt{2}\right)$
$=50\tan \left({\tan }^{-1}\frac{1}{2}+2\cdot \frac{\pi }{2}\right))+4\sqrt{2}\times \frac{1}{\sqrt{2}}$
$=50\left(\tan {\tan }^{-1}\frac{1}{2}\right)+4$
$=25+4=29$