Question

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

• A. $\frac{\mathrm{\pi }}{2}$
• B. $\frac{\mathrm{\pi }}{4}$
• C. ${\tan }^{-1}3$
• D. None of these

Answer 1

The correct option is B

$\frac{\mathrm{\pi }}{4}$

Explanation for the correct answer:

Finding the third angle:

The sum of all interior angles of a triangle is $\mathrm{\pi }$. then

$\mathrm{A}+\mathrm{B}+\mathrm{C}=\mathrm{\pi }$

Here, ${\tan }^{-1}2=\mathrm{A}$ and ${\tan }^{-1}3=\mathrm{B}$

Now,

$$\begin{gathered} C+{\tan }^{-1}2+{\tan }^{-1}3=\mathrm{\pi } \\ \Rightarrow \mathrm{C}+{\tan }^{-1}\left(\frac{2+3}{1-3\times 2}\right)=\mathrm{\pi }\left[\because {\tan }^{-1}x+{\tan }^{-1}y=\tan \left(\frac{x+y}{1-\mathrm{xy}}\right)\right] \\ \Rightarrow \mathrm{C}+{\tan }^{-1}\left(-1\right)=\mathrm{\pi } \\ \Rightarrow \mathbf{}\mathbf{}\mathrm{C}+{\tan }^{-1}\left(\tan \left(\mathrm{\pi }-\frac{\mathrm{\pi }}{4}\right)\right)=\mathrm{\pi } \\ \Rightarrow \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathrm{C}+\mathrm{\pi }-\frac{\mathrm{\pi }}{4}=\mathrm{\pi } \\ \Rightarrow \mathrm{C}=\frac{\mathrm{\pi }}{4} \end{gathered}$$

Hence, the correct option is B.