Question:
Mark the tick against the correct answer in the following:
$\tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{3}=?$
A. $\frac{\pi}{3}$
B. $\frac{\pi}{4}$
C. $\frac{\pi}{2}$
D. $\frac{2 \pi}{3}$
Solution:
To Find: The value of $\tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{3}$
Let, $x=\tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{3}$
Since we know that $\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right)$
$\Rightarrow \tan ^{-1} 1+\tan ^{-1} \frac{1}{3}=\tan ^{-1}\left(\frac{\frac{2}{2}+\frac{2}{3}}{1-\left(\frac{1}{3} \times \frac{1}{2}\right)}\right)=\tan ^{-1} 1=\frac{\pi}{4}$