Prove that:

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

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

Proof:

$\mathrm{LHS}=\tan ^{-1} \frac{2}{11}+\tan ^{-1} \frac{7}{24}$

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

$=\tan ^{-1}\left(\frac{48+77}{264-14}\right)$

$=\tan ^{-1} \frac{125}{250}$

$=\tan ^{-1} \frac{1}{2}$

$=\mathrm{RHS}$

Therefore LHS = RHS

Hence proved.