Question

Show that ${\tan }^{-1}\frac{2}{11}+{\tan }^{-1}\frac{7}{24}={\tan }^{-1}\frac{1}{2}$

Answer 1

Formula used:

${\tan }^{-1}x+{\tan }^{-1}y={\tan }^{-1}\left(\frac{x+y}{1-xy}\right)$

L.H.S$={\tan }^{-1}\left(\frac{2}{11}\right)+{\tan }^{-1}\left(\frac{7}{24}\right)$

$={\tan }^{-1}\left(\frac{\frac{2}{11}+\frac{7}{24}}{1-\frac{14}{24\times 11}}\right)$

$={\tan }^{-1}\left(\frac{48+77}{(24\times 11)-14}\right)$

$={\tan }^{-1}\left(\frac{125}{250}\right)$

$={\tan }^{-1}\left(\frac{1}{2}\right)=$R.H.S

Hence proved L.H.S=R.H.S