Let x and y be two acute angles such that x + y < 90° and sin(2x − 20°) = cos(2y + 20°). Under these conditions, what is the value of tan(x + y)?

Introduction / Context:
This problem is a trigonometric identity question involving acute angles and relationships between sine and cosine functions. It combines transformations of angles, the co-function identity between sine and cosine, and the use of angle sum restrictions to find the value of tan(x + y). Such questions are standard in trigonometry sections of aptitude tests and require both algebraic manipulation and a good grasp of basic trigonometric identities.

Given Data / Assumptions:

• x and y are acute angles (0° < x < 90°, 0° < y < 90°).
• x + y < 90°.
• sin(2x − 20°) = cos(2y + 20°).
• We need to determine tan(x + y).
• We assume the principal acute angle relationships, consistent with typical aptitude-style questions.

Concept / Approach:
The key idea is to use the co-function identity: cos θ = sin(90° − θ). The equation sin(2x − 20°) = cos(2y + 20°) can be rewritten entirely in terms of sine. Once we express cos(2y + 20°) as sin(90° − (2y + 20°)), we can equate the arguments of the sine functions. Under the standard principal-angle assumption for acute angles in such problems, we set the angles equal, solve for a relation between x and y, and then find the value of x + y and its tangent.

Step-by-Step Solution:
Start with sin(2x − 20°) = cos(2y + 20°).Use the co-function identity cos θ = sin(90° − θ).So cos(2y + 20°) = sin(90° − (2y + 20°)) = sin(70° − 2y).Thus sin(2x − 20°) = sin(70° − 2y).For acute-angle based aptitude questions, we typically take the principal solution, equating the arguments:2x − 20° = 70° − 2y.Rearrange: 2x + 2y = 90°.This gives x + y = 45°.Now tan(x + y) = tan(45°) = 1.

Verification / Alternative check:
We can choose a concrete pair of angles satisfying x + y = 45° and x, y acute, for example x = 25° and y = 20°. Then 2x − 20° = 2 * 25° − 20° = 30°, and 2y + 20° = 2 * 20° + 20° = 60°. We have sin(30°) = 1/2, cos(60°) = 1/2, so the given equation sin(2x − 20°) = cos(2y + 20°) holds. For this pair, x + y = 45°, and tan(45°) = 1, which is consistent with the derived answer. This kind of numerical test supports the algebraic reasoning.

Why Other Options Are Wrong:
√3 and 1/√3 correspond to tan(60°) and tan(30°) respectively, which would require x + y to be 60° or 30°, contradicting the equation obtained from the given relation. The option 2 + √2 does not correspond to a standard tangent value for simple angles and is not supported by any identity here. The option 0 would imply x + y = 0° or 180°, which is impossible for acute x and y with x + y < 90°. Only the value 1 matches tan(45°), which arises naturally from the relationship between x and y.

Common Pitfalls:
One frequent error is to forget to convert cos to sin using the co-function identity and instead attempt more complicated transformations. Another is to mishandle the equation sin A = sin B and ignore the principal acute-angle behaviour, leading to extraneous angle relations. Students may also incorrectly assume that x + y = 90°, which directly contradicts the given condition x + y < 90°. Careful attention to the identities and the conditions on the angles avoids these mistakes.

Final Answer:
The value of tan(x + y) under the given conditions is 1.