Simplify the trigonometric expression and choose its simplest equivalent form: sin 2A / (1 + cos 2A) Assume A is an angle for which the expression is defined.

Introduction / Context: This problem tests standard trigonometric identities and algebraic simplification. The expression sin 2A / (1 + cos 2A) is a classic form that reduces neatly to a single basic trig ratio.

Given Data / Assumptions:

• Expression: sin 2A / (1 + cos 2A) • Use standard identities for double angles • Assume denominator is not zero

Concept / Approach: Use: sin 2A = 2 sin A cos A cos 2A = cos^2 A − sin^2 A Also note: 1 + cos 2A = 1 + (cos^2 A − sin^2 A) = (sin^2 A + cos^2 A) + (cos^2 A − sin^2 A) = 2 cos^2 A. Then the expression becomes: (2 sin A cos A) / (2 cos^2 A) = sin A / cos A = tan A.

Step-by-Step Solution: 1) Replace sin 2A with 2 sin A cos A 2) Rewrite denominator: 1 + cos 2A = 1 + (cos^2 A − sin^2 A) 3) Use sin^2 A + cos^2 A = 1: 1 + cos 2A = (sin^2 A + cos^2 A) + (cos^2 A − sin^2 A) = 2 cos^2 A 4) Divide: (2 sin A cos A) / (2 cos^2 A) = sin A / cos A = tan A

Verification / Alternative check: Another common identity is: tan(A) = sin 2A / (1 + cos 2A), which you can also derive by multiplying numerator and denominator by (1 − cos 2A), but the double-angle substitution is the cleanest here.

Why Other Options Are Wrong: • cot A is the reciprocal of tan A, not equal to it. • sin A, cos A, sec A do not match the ratio sin A / cos A.

Common Pitfalls: • Forgetting that 1 + cos 2A becomes 2 cos^2 A. • Cancelling incorrectly and losing a factor of cos A.

Final Answer: tan A