Question

If ${\tan }^{-1}2,{\tan }^{-1}3$ are two angles of a triangle, then the third angle is

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{6}$
• D. $\frac{\pi }{2}$

Answer 1

The correct option is B $\frac{\pi }{4}$

Let the angles of triangle be
$A={\tan }^{-1}2,B={\tan }^{-1}3$ and angle $C$
We know that $A+B+C=\pi$
$C=\pi -({\tan }^{-1}2+{\tan }^{-1}3)\Rightarrow C=\pi -[\pi +{\tan }^{-1}\left(\frac{2+3}{1-2\cdot 3}\right)](\because xy=2\cdot 3=6>1)\Rightarrow C=\pi -\pi -{\tan }^{-1}(-1)=\frac{\pi }{4}$
$\therefore$ The third angle is $\frac{\pi }{4}$