In triangle ABC, if sin A = cos B, what is the value of cos C? Assume A, B, and C are the interior angles of a valid triangle and simplify using co-function identities.

Introduction / Context:
This problem connects triangle angle properties with co-function identities in trigonometry. In any triangle, the interior angles satisfy A + B + C = 180°. The condition sin A = cos B is a strong relationship because cosine can be rewritten as sine of a complementary angle: cos B = sin(90° − B). So sin A = sin(90° − B). For angles inside a triangle, A and B are between 0° and 180°, but in a standard (non-degenerate) triangle both A and B are positive and less than 180°. The most natural and intended solution is A = 90° − B, which implies A + B = 90°. Once A + B = 90°, the triangle sum gives C = 180° − 90° = 90°. Therefore cos C = cos 90° = 0. This is a classic aptitude outcome: a trig equality forces the triangle to be right-angled. The key is using identities plus the triangle angle sum, not numeric evaluation.

Given Data / Assumptions:

• In ΔABC: A + B + C = 180° • Given: sin A = cos B • Identity: cos B = sin(90° − B) • Required: cos C

Concept / Approach:
Rewrite cos B as sin(90° − B). Then sin A = sin(90° − B) implies A = 90° − B for the principal triangle-compatible case, giving A + B = 90°. Use the triangle sum to find C, then compute cos C exactly.

Step-by-Step Solution:
1) Start with sin A = cos B. 2) Convert cos to sin of a complement: cos B = sin(90° − B) 3) So the condition becomes: sin A = sin(90° − B) 4) For the intended triangle case, take: A = 90° − B 5) Then: A + B = 90° 6) Use triangle angle sum: C = 180° − (A + B) = 180° − 90° = 90° 7) Therefore: cos C = cos 90° = 0

Verification / Alternative check:
If C = 90°, the triangle is right-angled at C. Then A and B are complementary (A + B = 90°). In that situation, sin A = cos(90° − A) = cos B, which matches the given condition exactly. This confirms the reasoning is consistent with triangle geometry.

Why Other Options Are Wrong:
• 1, √3/2, 1/√2, 1/2: these are cosine values of angles 0°, 30°, 45°, and 60° respectively, but here C is forced to be 90°. • Any nonzero value would contradict the derived right angle at C.

Common Pitfalls:
• Forgetting cos B = sin(90° − B). • Ignoring the triangle sum A + B + C = 180°. • Overcomplicating by trying to assign numeric angles without using the identity.

Final Answer:
0