Question

Show that ${\tan }^{-1}1/2+{\tan }^{-1}2/11+{\tan }^{-1}4/3=\pi /2$

Answer 1

L.H.S

$={\tan }^{-1}\frac{1}{2}+{\tan }^{-1}\frac{2}{11}+{\tan }^{-1}\frac{4}{3}$

We know that

${\tan }^{-1}x+{\tan }^{-1}y+{\tan }^{-1}z={\tan }^{-1}\frac{(x+y+z)}{(1-xy-yz-zx)}$

Therefore,

$={\tan }^{-1}\left(\frac{\frac{1}{2}+\frac{2}{11}+\frac{4}{3}}{1-\frac{1}{2}\times \frac{2}{11}-\frac{2}{11}\times \frac{4}{3}-\frac{4}{3}\times \frac{1}{2}}\right)$

$={\tan }^{-1}\left(\frac{\frac{33+12+88}{66}}{1-\frac{1}{11}-\frac{8}{3\times 11}-\frac{2}{3}}\right)$

$={\tan }^{-1}\left(\frac{\frac{133}{66}}{1-\left(\frac{3+8+22}{3\times 11}\right)}\right)$

$={\tan }^{-1}\left(\frac{\frac{133}{66}}{1-\left(\frac{33}{33}\right)}\right)$

$={\tan }^{-1}\left(\frac{\frac{133}{66}}{1-1}\right)$

$={\tan }^{-1}\left(\frac{\frac{133}{66}}{0}\right)$

$={\tan }^{-1}\left(\infty \right)$

$={\tan }^{-1}\left(\tan \frac{\pi }{2}\right)$

$=\frac{\pi }{2}$

Hence, proved.