Inverse Trigonometric Functions

The result is driven by the addition formula for arctangents, which states that ${\tan }^{-1}x+{\tan }^{-1}y={\tan }^{-1}\frac{x+y}{1-xy}$ precisely when the product $xy<1$; the product condition matters because it keeps the combined angle within the principal branch and avoids the additive $\pi$ correction. Since both arctangents sum to the fixed angle $\frac{\pi }{4}$, applying the formula and then taking the tangent of both sides gives $\frac{x+y}{1-xy}=\tan \frac{\pi }{4}=1$. Cross-multiplying to clear the denominator yields $x+y=1-xy$, and collecting the product term on the left produces the symmetric algebraic relation $x+y+xy=1$. Option $x+y-xy=1$ flips the sign of the product term, which would correspond to a subtraction formula. Option $x+y=1$ omits the product term that the denominator contributes. Option $xy=1$ both violates the required $xy<1$ condition and contradicts the formula's hypothesis. This is the standard arctangent-sum identity tested throughout JEE Advanced. As a final plausibility check, choosing $x=y=\sqrt{2}-1\approx 0.414$ gives $x+y+xy\approx 0.828+0.172=1$, confirming the relation while keeping $xy\approx 0.172<1$ as required.