Inverse Trigonometric Functions

Question

Given that $	an^{-1} x + 	an^{-1} y = \frac{\pi}{4}$ holds for positive reals with $xy < 1$, the exact relationship connecting the two variables is which of the following?

RankGuru · Accepted Answer

The result is driven by the addition formula for arctangents, which states that $	an^{-1}x + 	an^{-1}y = 	an^{-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 $\frac{x + y}{1 - xy} = 	an\frac{\pi}{4} = 1$. 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.

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