Question

If ${\tan }^{-1}x-{\tan }^{-1}y={\tan }^{-1}A$, then $A=$

• A. $x-y$
• B. $x+y$
• C. $\frac{x-y}{1+xy}$
• D. $\frac{x+y}{1-xy}$

Answer 1

The correct option is C

$\frac{x-y}{1+xy}$

Explanation for the correct option:

Given equation: ${\tan }^{-1}x-{\tan }^{-1}y={\tan }^{-1}A$,

Use the formula ${\tan }^{-1}A-{\tan }^{-1}B={\tan }^{-1}\left(\frac{A-B}{1+AB}\right)$ in the above equation, we get

$\begin{array}{c}{\tan }^{-1}x-{\tan }^{-1}y={\tan }^{-1}A\\ \Rightarrow {\tan }^{-1}\frac{x-y}{1+xy}={\tan }^{-1}A\\ \Rightarrow \frac{x-y}{1+xy}=A\end{array}$

Hence, the correct option is (C).