Question

In \( \triangle ABC \), if \( (\sin A + \sin B)(\sin A - \sin B) = \sin C(\sin B + \sin C) \), then \( \angle A = \)

• A. \( 60^\circ \)
• B. \( 30^\circ \)
• C. \( 150^\circ \)
• D. \( 120^\circ \)

Hint:

For trigonometric equations involving angles in a triangle, look for symmetries or specific identities that can simplify the expression.

Answer 1

The Correct Option is D

Step 1: Simplify the given equation.
\[ (\sin A + \sin B)(\sin A - \sin B) = \sin^2 A - \sin^2 B \] \[ \sin C(\sin B + \sin C) = \sin C \sin B + \sin^2 C \]
Step 2: Equate the two expressions.
\[ \sin^2 A - \sin^2 B = \sin C \sin B + \sin^2 C \]
Step 3: Solve for angle \( A \).
Based on trigonometric properties and identities in triangles, we solve for \( \angle A = 120^\circ \).