For an acute angle A, simplify the trigonometric product (cosec A − sin A)(sec A − cos A)(tan A + cot A) and determine its exact value.

Introduction / Context:
This question explores the interplay between reciprocal trigonometric functions and basic identities. The product (cosec A − sin A)(sec A − cos A)(tan A + cot A) looks complicated, but after rewriting all functions in terms of sine and cosine, many cancellations occur. Simplification questions like this are used in aptitude and competitive exams to test fluency with trigonometric identities and algebraic manipulation.

Given Data / Assumptions:

• A is an acute angle, so sine, cosine, tangent, cotangent, secant and cosecant are all well defined and positive.
• The expression to simplify is (cosec A − sin A)(sec A − cos A)(tan A + cot A).
• We are allowed to use standard definitions cosec A = 1/sin A, sec A = 1/cos A, tan A = sin A/cos A, and cot A = cos A/sin A.
• The final answer should be a simple constant value.

Concept / Approach:
The main strategy is to express all trigonometric functions in terms of sin A and cos A. This converts the product into an algebraic expression in sin A and cos A, where we can factor and simplify. Often, terms like (1/sin A − sin A) and (1/cos A − cos A) give common factors. Together with tan A + cot A, which can be written as a single fraction, we can see cancellations that lead to a constant value.

Step-by-Step Solution:
Step 1: Rewrite cosec A − sin A as 1/sin A − sin A = (1 − sin^2 A)/sin A = cos^2 A/sin A.Step 2: Rewrite sec A − cos A as 1/cos A − cos A = (1 − cos^2 A)/cos A = sin^2 A/cos A.Step 3: Rewrite tan A + cot A as sin A/cos A + cos A/sin A = (sin^2 A + cos^2 A)/(sin A cos A) = 1/(sin A cos A), since sin^2 A + cos^2 A = 1.Step 4: Multiply the three factors: (cos^2 A/sin A) × (sin^2 A/cos A) × (1/(sin A cos A)).Step 5: Combine numerators and denominators: numerator = cos^2 A × sin^2 A × 1, denominator = sin A × cos A × sin A × cos A = sin^2 A cos^2 A.Step 6: Since numerator and denominator are both sin^2 A cos^2 A, the fraction simplifies to 1.Step 7: Therefore the entire product is equal to 1 for all acute angles A.

Verification / Alternative check:
Take a simple acute angle such as A = 45°. Then sin A = cos A = √2/2, tan A = 1, cot A = 1, sec A = √2 and cosec A = √2. Substitute into the original expression and compute numerically; the result comes out to 1, confirming our algebraic simplification. Any other acute angle will produce the same constant value when calculated correctly.

Why Other Options Are Wrong:
The options −1 and 0 are not possible because each individual factor is positive for an acute angle and the product cannot be negative or zero. The option 2 is also incorrect because our stepwise simplification leads precisely to 1, not a larger constant. The option involving cosec A sec A depends on A, but our final result is independent of the angle, so it cannot be correct.

Common Pitfalls:
Learners sometimes mis-handle the algebra when combining tan A and cot A, or they forget that sin^2 A + cos^2 A = 1. Another common mistake is failing to simplify 1/sin A − sin A and 1/cos A − cos A properly, leading to messy expressions that obscure the cancellations. Writing every function in terms of sin A and cos A and simplifying slowly prevents these errors.

Final Answer:
The simplified value of (cosec A − sin A)(sec A − cos A)(tan A + cot A) is 1.