Question

If tan^-1 x-tan^-1 y = $\frac{\mathrm{\pi }}{4}$, then x - y - xy = ____________________.

Answer 1

$$\begin{gathered} {\tan }^{-1}x-{\tan }^{-1}y=\frac{\mathrm{\pi }}{4} \\ \Rightarrow {\tan }^{-1}\left(\frac{x-y}{1+xy}\right)=\frac{\mathrm{\pi }}{4} \\ \Rightarrow \frac{x-y}{1+xy}=\tan \frac{\mathrm{\pi }}{4} \end{gathered}$$
$$\begin{gathered} \Rightarrow \frac{x-y}{1+xy}=1 \\ \Rightarrow x-y=1+xy \\ \Rightarrow x-y-xy=1 \end{gathered}$$

If tan⁻¹ x − tan⁻¹ y = $\frac{\mathrm{\pi }}{4}$, then x − y − xy = __1__.