If sec θ = cosec q for acute angles (0° < θ < 90° and 0° < q < 90°), find the exact value of: sin(θ + q) Use co-function identities and angle relationships for acute angles.

Introduction / Context: This question tests co-function relationships between trigonometric ratios and how they imply angle complementarity. Given sec θ = cosec q, you should convert both to reciprocals of basic functions: sec θ = 1/cos θ and cosec q = 1/sin q. So the equality implies cos θ = sin q. For acute angles, sin and cos values are positive and co-function identities apply cleanly. The key identity is: sin q = cos(90° − q). So cos θ = cos(90° − q). With both angles acute, the simplest and correct conclusion is θ = 90° − q, meaning θ and q are complementary. Once you know θ + q = 90°, the target becomes sin(90°), which is 1. This is a typical aptitude trick problem: it looks like it needs actual angle values, but it only needs identities and the acute-angle constraint to eliminate ambiguous solutions.

Given Data / Assumptions:

• sec θ = cosec q • 0° < θ < 90° and 0° < q < 90° • sec θ = 1/cos θ • cosec q = 1/sin q • Identity: sin q = cos(90° − q)

Concept / Approach: Convert the given equality into cos θ = sin q. Rewrite sin q as cos(90° − q), then equate cosine arguments under the acute-angle condition to obtain θ = 90° − q. Therefore θ + q = 90°, and sin(θ + q) = sin 90° = 1.

Step-by-Step Solution: 1) Start with sec θ = cosec q. 2) Convert to reciprocals: 1/cos θ = 1/sin q 3) Therefore: cos θ = sin q 4) Use co-function identity: sin q = cos(90° − q) 5) So: cos θ = cos(90° − q) 6) For acute angles, conclude: θ = 90° − q 7) Hence: θ + q = 90° 8) Evaluate: sin(θ + q) = sin 90° = 1

Verification / Alternative check: Pick a valid example: let q = 30°. Then sin q = 1/2, so cos θ must be 1/2, giving θ = 60° (acute). Now θ + q = 60° + 30° = 90°, and sin 90° = 1. This confirms the result works for a concrete acute-angle pair.

Why Other Options Are Wrong: • 0: would require θ + q = 0° or 180°, impossible for two positive acute angles. • 1/2: would require θ + q = 30° or 150°, not implied here. • 2 and 4: impossible because sine values lie between -1 and 1.

Common Pitfalls: • Not converting sec and cosec into cos and sin. • Forgetting the acute-angle condition and creating unnecessary alternate-angle cases. • Thinking you must find numeric values of θ and q instead of the relationship θ + q = 90°.

Final Answer: 1