For an acute angle θ, if cosec^2 θ = 625/576, find the exact simplified value of the expression (sin θ − cos θ) ÷ (sin θ + cos θ).

Introduction / Context:
This trigonometry question requires converting information about cosecant into values of sine and cosine, and then simplifying a ratio involving sin θ and cos θ. The given value of cosec^2 θ allows us to find sin θ exactly, and from there we can determine cos θ using the Pythagorean identity. Finally, substituting into (sin θ − cos θ)/(sin θ + cos θ) gives a rational number. Questions like this test precision and comfort with exact trigonometric values rather than approximate decimals.

Given Data / Assumptions:

• θ is an acute angle, so sin θ and cos θ are positive.
• cosec^2 θ = 625/576.
• cosec θ = 1/sin θ by definition.
• We must find (sin θ − cos θ)/(sin θ + cos θ) in exact fractional form.

Concept / Approach:
First, use the relation between cosecant and sine: cosec^2 θ = 1/sin^2 θ. From this we recover sin^2 θ and then sin θ, taking the positive root because θ is acute. Next, apply the identity sin^2 θ + cos^2 θ = 1 to find cos θ. With exact values for both functions, we substitute into the expression (sin θ − cos θ)/(sin θ + cos θ) and simplify the resulting fraction. All steps are done symbolically to avoid rounding errors.

Step-by-Step Solution:
Step 1: From cosec^2 θ = 625/576, we have 1/sin^2 θ = 625/576.Step 2: Invert both sides to find sin^2 θ: sin^2 θ = 576/625.Step 3: Since θ is acute, sin θ is positive, so sin θ = √(576/625) = 24/25.Step 4: Use the identity sin^2 θ + cos^2 θ = 1 to find cos^2 θ.Step 5: cos^2 θ = 1 − sin^2 θ = 1 − 576/625 = (625 − 576)/625 = 49/625.Step 6: Again, θ is acute, so cos θ is positive, giving cos θ = √(49/625) = 7/25.Step 7: Now compute sin θ − cos θ: 24/25 − 7/25 = 17/25.Step 8: Compute sin θ + cos θ: 24/25 + 7/25 = 31/25.Step 9: Form the required ratio: (sin θ − cos θ)/(sin θ + cos θ) = (17/25)/(31/25) = 17/31.

Verification / Alternative check:
We can verify the result numerically by approximating sin θ = 24/25 ≈ 0.96 and cos θ = 7/25 ≈ 0.28. Then sin θ − cos θ ≈ 0.68, sin θ + cos θ ≈ 1.24, and their ratio is approximately 0.548. The fraction 17/31 is roughly 0.548 as well, so the numerical check aligns with the exact algebraic answer. This reinforces the correctness of the exact fraction 17/31.

Why Other Options Are Wrong:
The value 1 would require sin θ = cos θ, which is not consistent with sin θ = 24/25 and cos θ = 7/25. The option 31/17 is simply the reciprocal of the correct answer and arises if one accidentally flips numerator and denominator. The fraction 14/25 does not correspond to any ratio formed from 24/25 and 7/25 in this expression. The additional option 7/25 is equal to cos θ and does not represent the required ratio. Only 17/31 matches the correctly simplified value of the expression.

Common Pitfalls:
Frequent mistakes include forgetting that cosec θ is 1/sin θ and treating the given number directly as sin^2 θ instead of its reciprocal, or taking the negative square root even though θ is acute. Another issue is mismanaging fractions when simplifying (sin θ − cos θ)/(sin θ + cos θ), for example by not cancelling the common denominator 25 correctly. Careful fraction operations and attention to the sign and quadrants of trigonometric functions avoid these errors.

Final Answer:
The simplified value of (sin θ − cos θ)/(sin θ + cos θ) is 17/31.