Question:
Mark the tick against the correct answer in the following:
$2 \tan ^{-1} \frac{1}{3}=?$
A. $\tan ^{-1} \frac{3}{2}$
B. $\tan ^{-1} \frac{3}{4}$
C. $\tan ^{-1} \frac{4}{3}$
D. none of these
Solution:
To Find: The value of $2 \tan ^{-1} \frac{1}{3}$ i.e, $\tan ^{-1} \frac{1}{3}+\tan ^{-1} \frac{1}{3}$
Let, $x=\tan ^{-1} \frac{1}{3}+\tan ^{-1} \frac{1}{3}$
Since we know that $\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right)$
$\Rightarrow \tan ^{-1} 1+\tan ^{-1} \frac{1}{3}=\tan ^{-1}\left(\frac{\frac{1}{3}+\frac{2}{3}}{1-\left(\frac{1}{3} \times \frac{1}{3}\right)}\right)=\tan ^{-1} \frac{3}{4}$
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