Question

If tan⁻¹ x + tan⁻¹ y = `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,
\[\tan^{- 1} x + \tan^{- 1} y = \frac{\pi}{4}\]
\[ \Rightarrow \tan^{- 1} \left( \frac{x + y}{1 - xy} \right) = \frac{\pi}{4}\]
\[ \Rightarrow \frac{x + y}{1 - xy} = \tan\frac{\pi}{4}\]
\[ \Rightarrow \frac{x + y}{1 - xy} = 1 \]
\[ \Rightarrow x + y = 1 - xy\]
\[ \Rightarrow x + y + xy = 1\]
∴ \[x + y + xy = 1\]