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