If cosec θ + sin θ = 5/2 for an acute angle (0° < θ < 90°), then what is the value of: cosec θ − sin θ Find the exact value using algebraic simplification.

Introduction / Context:
This question tests how to work with reciprocal trigonometric functions using algebra. The expression involves cosec θ (which equals 1/sin θ) and sin θ itself. A standard technique is to let sin θ = s (with 0 < s < 1 for an acute angle) and rewrite cosec θ as 1/s. Then the given equation becomes an algebraic equation in s. Once s is found, you can compute cosec θ − sin θ = 1/s − s. Another neat shortcut is to use the identity: (cosec θ + sin θ)(cosec θ − sin θ) = cosec^2 θ − sin^2 θ. But because cosec^2 θ = 1/sin^2 θ, it often becomes simplest to solve for s directly first. The acute-angle condition ensures sin θ is positive, which avoids sign confusion. The final answer should be a rational number in this problem due to clean cancellation.

Given Data / Assumptions:

• cosec θ + sin θ = 5/2 • 0° < θ < 90° so sin θ > 0 • cosec θ = 1 / sin θ • Required: cosec θ − sin θ

Concept / Approach:
Let s = sin θ. Then cosec θ = 1/s. The given becomes: 1/s + s = 5/2. Solve this quadratic for s using simple factoring. Choose the valid root that lies between 0 and 1. Then compute 1/s − s to get the requested value.

Step-by-Step Solution:
1) Let s = sin θ (so 0 < s < 1). 2) Then cosec θ = 1/s. 3) Given equation: 1/s + s = 5/2 4) Multiply both sides by 2s: 2 + 2s^2 = 5s 5) Rearrange: 2s^2 − 5s + 2 = 0 6) Factor: (2s − 1)(s − 2) = 0 7) Solutions: s = 1/2 or s = 2. Since 0 < s < 1, take s = 1/2. 8) Compute cosec θ − sin θ: 1/s − s = 1/(1/2) − 1/2 = 2 − 1/2 = 3/2

Verification / Alternative check:
If sin θ = 1/2, then θ = 30° (acute), and cosec θ = 2. Then cosec θ + sin θ = 2 + 1/2 = 5/2, which matches exactly. Therefore the computed difference 2 − 1/2 = 3/2 is correct.

Why Other Options Are Wrong:
• -3/2: wrong sign; both cosec θ and sin θ are positive here and cosec θ > sin θ. • 1/2: would mean 1/s − s = 1/2, not true for s = 1/2. • √3/2 or -√3/2: unrelated surd forms; the algebra forces a rational result here.

Common Pitfalls:
• Forgetting cosec θ = 1/sin θ. • Keeping the invalid root s = 2, which is impossible for sine. • Sign confusion despite the acute-angle constraint.

Final Answer:
3/2