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

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