Question

The third angle of the triangle, if its two angles are ${\tan }^{-1}2\mathrm{and}{\tan }^{-1}3$ is

• A. $\frac{3\pi }{4}$
• B. $\frac{\pi }{4}$
• C. ${\tan }^{-1}4$
• D. None.

Answer 1

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

We know that the sum of the angles of a triangle is $\pi .$
And ${\tan }^{-1}x+{\tan }^{-1}y=\pi +{\tan }^{-1}\left(\frac{x+y}{1-\mathrm{xy}}\right),\mathrm{if}x>0,y>0\mathrm{and}\mathrm{xy}>1.$

$\mathrm{Let}\mathrm{the}\mathrm{third}\mathrm{angle}\mathrm{be}x.\therefore {\tan }^{-1}2+{\tan }^{-1}3+x=\pi \Rightarrow x=\pi -\left({\tan }^{-1}2+{\tan }^{-1}3\right)\Rightarrow x=\pi -\left(\pi +{\tan }^{-1}\left(\frac{2+3}{1-2\times 3}\right)\right]\Rightarrow x=\pi -\left(\pi +{\tan }^{-1}\left(-1\right)\right)\Rightarrow x=\pi -\left(\pi -\frac{\pi }{4}\right)\Rightarrow x=\frac{\pi }{4}$