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

Introduction:
This trigonometry question tests your understanding of angle relationships and basic trigonometric identities. You are given a relation between sine and cosine involving transformed angles and asked to find the tangent of a sum of angles, tan(x + y). Recognising how to convert cosine into sine and then solving the resulting equation is the core idea here.

Given Data / Assumptions:

• x and y are acute angles, so 0° < x < 90° and 0° < y < 90°.
• x + y < 90°.
• sin(2x − 20°) = cos(2y + 20°).
• We are required to find tan(x + y).

Concept / Approach:
We use the identity cos A = sin(90° − A) to convert the cosine term to a sine term. This lets us equate arguments of sine by using the fact that sin θ = sin φ implies either θ = φ + 360°k or θ = 180° − φ + 360°k. Under the given acute angle conditions and the restriction x + y < 90°, we select the valid relation and compute x + y, then evaluate tan(x + y).

Step-by-Step Solution:
Rewrite cos(2y + 20°) as a sine function:cos(2y + 20°) = sin(90° − (2y + 20°)) = sin(70° − 2y).So the equation becomes sin(2x − 20°) = sin(70° − 2y).For principal values, one possibility is 2x − 20° = 70° − 2y.Then 2x + 2y = 90° ⇒ x + y = 45°.The alternative solution 2x − 20° = 180° − (70° − 2y) = 110° + 2y would lead to x − y = 65°, which combined with x and y acute gives x + y > 90°, contradicting the condition x + y < 90°. So we reject that.Thus x + y = 45°.Now tan(x + y) = tan 45° = 1.

Verification / Alternative check:
You can choose specific values satisfying x + y = 45° and the conditions, for example x = 30°, y = 15°. Then 2x − 20° = 40°, 2y + 20° = 50°. We have sin 40° and cos 50°, and since cos 50° = sin(40°), the equation holds. For these x and y, x + y = 45°, and tan 45° = 1, confirming our result.

Why Other Options Are Wrong:
√3 and 1/√3 correspond to 60° and 30° tangents, which would require x + y to be 60° or 30°, not 45°. The value 2 + √2 does not match any standard simple angle tangent. Zero would require x + y to be 0° or 180°, which is impossible here since x and y are acute and their sum is less than 90°.

Common Pitfalls:
Common errors include forgetting the identity cos A = sin(90° − A), mixing up the general solutions of sin θ = sin φ, or accepting the second sine solution without checking it against the given inequality x + y < 90°. Always verify which branch of the trigonometric equation is compatible with the constraints.

Final Answer:
The required value of tan(x + y) is 1.