Question

Find the value of $\tan [{\cos }^{-1}\left(\frac{4}{5}\right)+{\tan }^{-1}\left(\frac{2}{3}\right)]$

Answer 1

Given, $\tan [{\cos }^{-1}\left(\frac{4}{5}\right)+{\tan }^{-1}\left(\frac{2}{3}\right)]=\tan \left[{\tan }^{-1}\left(\frac{\sqrt{5^2-4^2}}{4}\right)+{\tan }^{-1}\left(\frac{2}{3}\right)\right]$
$=\tan [{\tan }^{-1}\left(\frac{3}{4}\right)+{\tan }^{-1}\left(\frac{2}{3}\right)]=\tan \left[{\tan }^{-1}\left(\frac{\frac{3}{4}+\frac{2}{3}}{1-\frac{3}{4}\times \frac{2}{3}}\right)\right]$
$=\tan \left[{\tan }^{-1}\left(\frac{9+8}{12-6}\right)\right]=\tan \left[{\tan }^{-1}\left(\frac{17}{6}\right)\right]=\frac{17}{6}$