If $\tan ^{-1} 2, \tan ^{-1} 3$ are measures of two angles of triangle, then the measure of its third angle is________________.
Let the measure of third angle of the triangle be x.
Now,
$\tan ^{-1} 2+\tan ^{-1} 3+x=\pi$ (Angle sum property of triangle)
$\Rightarrow \pi+\tan ^{-1}\left(\frac{2+3}{1-2 \times 3}\right)+x=\pi$ $\left[\tan ^{-1} x+\tan ^{-1} y=\pi+\tan \left(\frac{x+y}{1-x y}\right)\right.$, if $\left.x y>1\right]$
$\Rightarrow \tan ^{-1}(-1)+x=0$
$\Rightarrow-\frac{\pi}{4}+x=0$
$\Rightarrow x=\frac{\pi}{4}$
Thus, the measure of third angle of the triangle is $\frac{\pi}{4}$.
If $\tan ^{-1} 2, \tan ^{-1} 3$ are measures of two angles of triangle, then the measure of its third angle is $\frac{\pi}{4}$