Question

Prove: ${\tan }^{-1}\frac{2}{11}+{\tan }^{-1}\frac{7}{24}={\tan }^{-1}\frac{1}{2}$

Answer 1

$\text{L.H.S.}={\tan }^{-1}\frac{2}{11}+{\tan }^{-1}\frac{7}{24}$

$={\tan }^{-1}\frac{\frac{2}{11}+\frac{7}{24}}{1-\frac{2}{11}.\frac{7}{24}}$, .....$[\because {\tan }^{-1}x+{\tan }^{-1}y={\tan }^{-1}\frac{x+y}{1-xy}]$

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

$={\tan }^{-1}\frac{48+77}{264-14}={\tan }^{-1}\frac{125}{250}={\tan }^{-1}\frac{1}{2}$

$=\text{R.H.S.}$