If $\tan^{-1} x + \tan^{-1} y = \frac{\pi}{4},$ then compute $xy + x + y.$
I recall that $\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta},$ so I can use this to simplify the given equation. Taking the tangent of both sides, I get $\tan(\tan^{-1} x + \tan^{-1} y) = \tan \frac{\pi}{4}.$ Using the sum formula, I get $\frac{x + y}{1 - xy} = 1.$ Cross-multiplying, I get $x + y = 1 - xy.$ I want to find $xy + x + y,$ so I can add $xy$ to both sides and get $xy + x + y = 1.$ 