Question

For $x=2$and $y=3$ the angle ${\tan }^{-1}\left(x\right)+{\tan }^{-1}\left(y\right)$ equals

• A. $135°$
• B. $45°$
• C. $90°$
• D. $180°$

Answer 1

The correct option is A

$135°$

Explanation of the correct option :

Find the angle ${\tan }^{-1}\left(x\right)+{\tan }^{-1}\left(y\right)$

Given that $x=2$and $y=3$

As $x>1,y>1$$\Rightarrow xy>1$

We know that $\begin{array}{rcl}{\tan }^{-1}\left(x\right)+\mathrm{ta}{n}^{-1}\left(y\right)& =& \mathrm{\pi }+{\tan }^{-1}\left[\frac{(\mathrm{x}+\mathrm{y})}{1-\mathrm{xy}}\right]\end{array}$for $\begin{array}{rcl}xy& >& 1\end{array}$

Now, put $x=2$and $y=3$ we get

$\begin{array}{rcl}{\tan }^{-1}\left(2\right)+{\tan }^{-1}\left(3\right)& =& \mathrm{\pi }+{\tan }^{-1}\left[\frac{(2+3)}{1-6}\right]\\ & =& \mathrm{\pi }+{\tan }^{-1}\left[\frac{5}{-5}\right]\\ & =& \mathrm{\pi }+{\tan }^{-1}\left[-1\right]\\ & =& \mathrm{\pi }-\frac{\mathrm{\pi }}{4}\\ & =& \frac{3\mathrm{\pi }}{4}\\ & =& \frac{3\times 180°}{4}\\ & =& 135°\end{array}$

Hence, the correct option is A.