Question

If \[ \frac{\sin(A + B)}{\sin(A - B)} = \frac{\cos(C + D)}{\cos(C - D)}, \] then \( \tan A \cot B = \)

• A. \( \cot C \cot D \)
• B. \( -\tan C \tan D \)
• C. \( \tan C \tan D \)
• D. \( -\cot C \cot D \)

Hint:

Use trigonometric identities for sums and differences to simplify complex expressions. Look for patterns that match known identities.

Answer 1

The Correct Option is D

Step 1: Use trigonometric identities.
We are given the equation: \[ \frac{\sin(A + B)}{\sin(A - B)} = \frac{\cos(C + D)}{\cos(C - D)}. \] Using trigonometric identities for the sine and cosine of sums and differences, we simplify both sides of the equation.
Step 2: Simplify and solve.
After simplifying, we find that \( \tan A \cot B = -\cot C \cot D \). Thus, the correct answer is option (D).