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Question
If tan−1 x + tan−1 y = `pi/4`, then write the value of x + y + xy.
Solution
We know that
\[\tan^{- 1} x + \tan^{- 1} y = \tan^{- 1} \left( \frac{x + y}{1 - xy} \right)\]
Now,
\[\tan^{- 1} x + \tan^{- 1} y = \frac{\pi}{4}\]
\[ \Rightarrow \tan^{- 1} \left( \frac{x + y}{1 - xy} \right) = \frac{\pi}{4}\]
\[ \Rightarrow \frac{x + y}{1 - xy} = \tan\frac{\pi}{4}\]
\[ \Rightarrow \frac{x + y}{1 - xy} = 1 \]
\[ \Rightarrow x + y = 1 - xy\]
\[ \Rightarrow x + y + xy = 1\]
∴ \[x + y + xy = 1\]
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