Prove that ^ -1 2 3 =1 2 ^ -1 12 5 - Vidyadip

Question

Prove that $\tan ^{-1} \frac{2}{3}=\frac{1}{2} \tan ^{-1} \frac{12}{5}$

✓

Answer

$\text{L.H.S.}=\tan ^{-1} \frac{2}{3}= A ($suppose$)$
or
$\tan A=\frac{2}{3}$
$\tan 2 A=\frac{2 \tan A}{1-\tan ^2 A}$
$=\frac{2 \times \frac{2}{3}}{1-\left(\frac{2}{3}\right)^2}=\frac{\frac{4}{3}}{1-\frac{4}{9}}=\frac{\frac{4}{3}}{\frac{5}{9}}$
$\Rightarrow \quad \tan 2 A=\frac{4}{3} \times \frac{9}{5}=\frac{12}{5}$
$\Rightarrow \quad 2 A=\tan ^{-1}\left(\frac{12}{5}\right)$
$\Rightarrow \quad A =\frac{1}{2} \tan ^{-1}\left(\frac{12}{5}\right)$
$\Rightarrow \quad \tan ^{-1}\left(\frac{2}{3}\right)=\frac{1}{2} \tan ^{-1}\left(\frac{12}{5}\right) \quad$
Hence proved.

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