Question

If \(\sin A + \cos B = a\) and \(\sin B + \cos A = b\), then \(\sin(A + B)\) is equal to

• A. \(\frac{a^2 + b^2}{2}\)
• B. \(\frac{a^2 - b^2 + 2}{2}\)
• C. \(\frac{a^2 + b^2 - 2}{2}\)
• D. None of these

Hint:

\(\sin A \cos B + \sin B \cos A = \sin(A + B)\). Square and add to eliminate other terms.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Square both given equations and add.

Step 2: Detailed Explanation:
\(a^2 = \sin^2 A + \cos^2 B + 2\sin A \cos B\)
\(b^2 = \sin^2 B + \cos^2 A + 2\sin B \cos A\)
Add: \(a^2 + b^2 = (\sin^2 A + \cos^2 A) + (\sin^2 B + \cos^2 B) + 2(\sin A \cos B + \sin B \cos A)\)
\(a^2 + b^2 = 1 + 1 + 2\sin(A + B) = 2 + 2\sin(A + B)\)
\(\implies \sin(A + B) = \frac{a^2 + b^2 - 2}{2}\)

Step 3: Final Answer:
Option (C).