Prove that:

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

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

Proof:

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

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

$=\tan ^{-1}\left(\frac{11+4}{22-2}\right)$

$=\tan ^{-1} \frac{15}{20}$

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

$=\mathrm{RHS}$

Therefore LHS = RHS

Hence proved.