Question

If tan⁻¹ x + tan⁻¹ y = $\frac{\mathrm{\pi }}{4}$, then write the value of x + y + xy.

Answer 1

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

∴ $x+y+xy=1$