Simplify the expression cos(90° - B) sin(C - A) + sin(90° + A) cos(B + C) - sin(90° - C) cos(A + B) and find its value.

Introduction / Context:
This trigonometric simplification problem involves several angles expressed in terms of A, B and C together with 90 degrees. It is designed to test your familiarity with cofunction identities such as cos(90° - θ) and sin(90° ± θ), as well as your ability to expand products using sine and cosine addition formulas and then combine terms systematically.

Given Data / Assumptions:

• Expression: cos(90° - B) sin(C - A) + sin(90° + A) cos(B + C) - sin(90° - C) cos(A + B).
• Angles are in degrees.
• A, B and C are real angles for which the functions are defined.

Concept / Approach:
We first convert functions of 90° ± angle into sine or cosine of the original angle using identities:
cos(90° - B) = sin B
sin(90° + A) = cos A
sin(90° - C) = cos C
After these substitutions, each term involves products like sin B sin(C - A) or cos A cos(B + C). Then we use sine and cosine of difference and sum identities to expand sin(C - A), cos(B + C) and cos(A + B), and look for cancellation between similar terms.

Step-by-Step Solution:
Step 1: Rewrite cos(90° - B) as sin B. Step 2: Rewrite sin(90° + A) as cos A. Step 3: Rewrite sin(90° - C) as cos C. Step 4: The expression becomes sin B sin(C - A) + cos A cos(B + C) - cos C cos(A + B). Step 5: Use sin(C - A) = sin C cos A - cos C sin A to expand the first product. Step 6: The first term becomes sin B[sin C cos A - cos C sin A] = sin B sin C cos A - sin B cos C sin A. Step 7: Expand cos(B + C) using cos(B + C) = cos B cos C - sin B sin C. The second term becomes cos A[cos B cos C - sin B sin C] = cos A cos B cos C - cos A sin B sin C. Step 8: Expand cos(A + B) using cos(A + B) = cos A cos B - sin A sin B. The third term is -cos C[cos A cos B - sin A sin B] = -cos C cos A cos B + cos C sin A sin B. Step 9: Collect all terms: sin B sin C cos A - sin B cos C sin A + cos A cos B cos C - cos A sin B sin C - cos C cos A cos B + cos C sin A sin B. Step 10: Pair and cancel like terms. The terms sin B sin C cos A and -cos A sin B sin C cancel. The terms cos A cos B cos C and -cos C cos A cos B cancel. The terms -sin B cos C sin A and cos C sin A sin B also cancel. Step 11: After all cancellations, no terms remain, so the expression equals 0.

Verification / Alternative check:
You can verify the result numerically by choosing specific values for A, B and C, such as A = 30°, B = 40° and C = 50°, and evaluating the original expression with a calculator. The numerical sum will be extremely close to zero, confirming that the algebraic simplification is correct.

Why Other Options Are Wrong:
1, sin(A + B - C), cos(B + C - A) and cos(A + B + C) all represent non zero trigonometric expressions in general and do not match the complete cancellation observed. They might look plausible as simplified forms, but direct expansion and cancellation show that the entire expression reduces exactly to 0 for all valid A, B and C.

Common Pitfalls:
Mistakes often occur from incorrect signs in the expansions of sin(C - A) or cos(A + B). It is also easy to overlook that all terms cancel if not grouped carefully. Writing each expanded term clearly and checking for matching positive and negative pairs is crucial to reach the final simplified form.

Final Answer:
The value of the given expression is 0.