Simplify the trigonometric-algebraic expression: (1 − sin A cos A)(sin A + cos A) Choose the simplest equivalent expression (assume the expression is defined for the given angle A).

Introduction / Context: This question tests identity recognition. Expressions that contain (sin A + cos A) often combine nicely with sin^2 A + cos^2 A = 1. A common target form is sin^3 A + cos^3 A, which can be factorized as (sin A + cos A)(sin^2 A − sin A cos A + cos^2 A).

Given Data / Assumptions:

• Expression: (1 − sin A cos A)(sin A + cos A) • Identity: sin^2 A + cos^2 A = 1 • Required: simplest equivalent form

Concept / Approach: Rewrite the factor (1 − sin A cos A) using sin^2 A + cos^2 A: 1 − sin A cos A = (sin^2 A + cos^2 A) − sin A cos A = sin^2 A − sin A cos A + cos^2 A. Then use the known factorization: u^3 + v^3 = (u + v)(u^2 − uv + v^2), with u = sin A and v = cos A.

Step-by-Step Solution: 1) Start with: (1 − sin A cos A)(sin A + cos A) 2) Replace 1 by sin^2 A + cos^2 A: 1 − sin A cos A = sin^2 A + cos^2 A − sin A cos A 3) Rearrange: = sin^2 A − sin A cos A + cos^2 A 4) So the full expression becomes: (sin A + cos A)(sin^2 A − sin A cos A + cos^2 A) 5) Recognize as u^3 + v^3 factorization: = sin^3 A + cos^3 A

Verification / Alternative check: Take A = 0°: sin A = 0, cos A = 1. Original: (1 − 0)(0 + 1) = 1. Right side: sin^3 A + cos^3 A = 0 + 1 = 1. Matches. This confirms the identity-based simplification.

Why Other Options Are Wrong: • sin^2 A − cos^2 A is unrelated to the cube-sum structure. • 0 or 1 cannot be identically equal for all A (depends on A). • sin A + cos A misses the extra factor.

Common Pitfalls: • Expanding blindly and not using sin^2 A + cos^2 A = 1. • Forgetting the identity for u^3 + v^3 factorization.

Final Answer: sin^3 A + cos^3 A