Question

Two angles of a triangle are ${\cot }^{-1}2$ and ${\cot }^{-1}3$. Then the third angle

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

Answer 1

Answer: (B)

$\frac{3\pi }{4}$

The sum of three angles of triangle is $\pi$

$\Rightarrow {\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{1}{2}+{\tan }^{-1}x=\pi$

$\Rightarrow {\tan }^{-1}\frac{(\frac{1}{3}+\frac{1}{2})}{1-\frac{1}{6}}+{\tan }^{-1}x=\pi$

$\Rightarrow {\tan }^{-1}\left(\frac{5}{5}\right)+{\tan }^{-1}x=\pi$

$\Rightarrow {\tan }^{-1}x=\pi -\frac{\pi }{4}=\frac{3\pi }{4}$.