Consider the trigonometric product (sin A − cosec A)(sec A − cos A)(tan A + cot A) for an angle A where all these functions are defined. What is the simplified numerical value of this entire expression?

Introduction / Context:
This problem involves simplifying a product of several trigonometric expressions. It tests understanding of reciprocal relationships between sine, cosine, and their reciprocals, along with how tangent and cotangent can be rewritten in terms of sine and cosine. The final result turns out to be a constant, independent of the specific angle A, as long as the functions are defined.

Given Data / Assumptions:

• The expression is (sin A − cosec A)(sec A − cos A)(tan A + cot A).
• Angle A is such that sin A, cos A, sec A, cosec A, tan A, and cot A are all defined and non zero where needed.
• We must simplify this expression to a single numerical value if possible.

Concept / Approach:
Rewrite all trigonometric functions in terms of sine and cosine. Use the reciprocal relations cosec A = 1/sin A, sec A = 1/cos A, tan A = sin A / cos A, and cot A = cos A / sin A. Simplify each bracket separately to express them as algebraic fractions in sin A and cos A. Then multiply them to see if massive cancellation occurs, leading to a constant value.

Step-by-Step Solution:
First simplify sin A − cosec A: this is sin A − 1/sin A = (sin^2 A − 1)/sin A = −(1 − sin^2 A)/sin A = −cos^2 A / sin A.Next simplify sec A − cos A: this is 1/cos A − cos A = (1 − cos^2 A)/cos A = sin^2 A / cos A.Simplify tan A + cot A: this is (sin A / cos A) + (cos A / sin A) = (sin^2 A + cos^2 A)/(sin A cos A) = 1/(sin A cos A).Multiply the first two results: (−cos^2 A / sin A) * (sin^2 A / cos A) = −(cos^2 A sin^2 A)/(sin A cos A) = −sin A cos A.Now multiply this with the third factor: (−sin A cos A) * (1/(sin A cos A)) = −1.Thus the entire expression simplifies to the constant value −1.

Verification / Alternative check:
Choose a specific angle A where all functions are defined, for example A = 45 degrees. Compute each term numerically using a calculator: sin 45° = cos 45° = sqrt(2)/2, so the simplified step by step expression should give a result very close to −1. Carrying out this numerical evaluation confirms the algebraic simplification and shows the expression is indeed constant for all valid A.

Why Other Options Are Wrong:

• 1 and 2 would require different sign structures or incomplete cancellations.
• 0 would require one of the factors to be zero, but none of the simplified factors is zero for a general valid A.
• −2 does not appear naturally from the algebraic manipulations of the expression.
• Only −1 matches the exact result from the complete simplification.

Common Pitfalls:

• Making algebraic sign errors when rewriting sin A − cosec A and sec A − cos A.
• Incorrectly adding tan A and cot A, for example forgetting to combine over a common denominator.
• Failing to recognise that massive cancellation happens, and stopping at a more complicated intermediate expression rather than the constant −1.

Final Answer:
-1