Question:
Mark the tick against the correct answer in the following:
$\left(\tan ^{-1} 2+\tan ^{-1} 3\right)=?$
A. $\frac{-\pi}{4}$
B. $\frac{\pi}{4}$
C. $\frac{3 \pi}{4}$
D. $\pi$
Solution:
To Find: The value of $\tan ^{-1} 2+\tan ^{-1} 3$
Since we know that $\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right)$
$\Rightarrow \tan ^{-1} 2+\tan ^{-1} 3=\tan ^{-1}\left(\frac{2+3}{1-(2 \times 3)}\right)$
$=\tan ^{-1}\left(\frac{5}{-5}\right)$
$=\tan ^{-1}(-1)$
Since the principle value of tan lies in the range $[0, \pi]$
$\Rightarrow \tan ^{-1}(-1)=\frac{3 \pi}{4}$