Question:
The value of $\tan ^{-1} 2+\tan ^{-1} 3$ is___________________.
Solution:
We know
$\tan ^{-1} x+\tan ^{-1} y=\pi+\tan ^{-1}\left(\frac{x+y}{1-x y}\right)$, if $x y>1$
$\therefore \tan ^{-1} 2+\tan ^{-1} 3$
$=\pi+\tan ^{-1}\left(\frac{2+3}{1-2 \times 3}\right)$
$=\pi+\tan ^{-1}(-1)$
$=\pi-\frac{\pi}{4}$
$=\frac{3 \pi}{4}$
The value of $\tan ^{-1} 2+\tan ^{-1} 3$ is
$\frac{3 \pi}{4}$