Simplify the trigonometric expression [sin(90° - A) + cos(180° - 2A)] / [cos(90° - 2A) + sin(180° - A)] and express it in terms of A as a single trigonometric function.

Introduction / Context:
This problem asks you to simplify a trigonometric expression involving angles of the form 90° - A and 180° - 2A or 180° - A. These are standard complementary and supplementary angle forms. The aim is to write the entire fraction as a simple function of A, most likely involving a half angle. This type of question tests knowledge of basic identities and the ability to recognize patterns that reduce to tan(A / 2) or similar functions.

Given Data / Assumptions:

• Expression: E = [sin(90° - A) + cos(180° - 2A)] / [cos(90° - 2A) + sin(180° - A)].
• All angles are in degrees.
• A is such that all trigonometric functions involved are defined.
• Standard cofunction and supplementary angle identities hold.

Concept / Approach:
We use the identities: sin(90° - A) = cos A, cos(180° - 2A) = -cos 2A, cos(90° - 2A) = sin 2A, sin(180° - A) = sin A. By substituting these into the expression, we rewrite it purely in terms of sin A, cos A, sin 2A and cos 2A. Then we simplify using double angle identities and finally recognize the result as a half angle tangent function tan(A / 2).

Step-by-Step Solution:
Step 1: Replace each term using cofunction and supplementary angle identities: sin(90° - A) = cos A, cos(180° - 2A) = -cos 2A, cos(90° - 2A) = sin 2A, sin(180° - A) = sin A. Step 2: Substitute in the numerator and denominator: Numerator N = cos A - cos 2A. Denominator D = sin 2A + sin A. Step 3: Express cos 2A and sin 2A in terms of sin A and cos A: cos 2A = cos^2 A - sin^2 A, sin 2A = 2 sin A cos A. Step 4: Replace and simplify algebraically to express N and D in terms of sin A and cos A and then simplify the ratio. Step 5: After simplification, the fraction reduces to a known half angle form, which equals tan(A / 2).

Verification / Alternative check:
To confirm, pick a numerical value such as A = 60°. Compute each term directly: sin(90° - 60°), cos(180° - 120°), cos(90° - 120°), and sin(180° - 60°). Substitute into the expression and evaluate the fraction in degree mode. Then compute tan(A / 2) = tan 30°. Both values will match numerically. Since trigonometric identities are exact, one non trivial verification strongly supports that the simplified form is tan(A / 2).

Why Other Options Are Wrong:
Option a, sin(A / 2) cos A, and option d, sin A cos(A / 2), are products rather than ratios and do not naturally arise from this type of fraction. Option b, cot(A / 2), is the reciprocal of the correct answer and would require the numerator and denominator to be swapped. Option e, cos A sin(A / 2), also has the wrong structure. Only option c, tan(A / 2), matches the simplified result obtained from systematic use of the identities.

Common Pitfalls:
Students often misapply the identities for cos(180° - 2A) or cos(90° - 2A), sometimes dropping the minus sign or confusing sine and cosine. Another frequent problem is attempting to simplify too quickly without systematically expressing everything in sin A and cos A. Errors in double angle formulas can also lead to incorrect conclusions. Writing each identity clearly and proceeding in small algebraic steps reduces the risk of mistakes.

Final Answer:
Hence, the expression simplifies to tan(A / 2).