Question

In a triangle ABC, if \[ \frac{\sin A - \sin C}{\cos C - \cos A} = \cot B, \text{ then A, B, C are in} \]

• A. Arithmetic - Geometric progression
• B. Harmonic Progression
• C. Geometric progression
• D. Arithmetic progression

Hint:

When given a trigonometric identity involving the angles of a triangle, use algebraic manipulations to find the type of progression between the angles.

Answer 1

The Correct Option is D

Step 1: Analyzing the given equation.
We are given the equation: \[ \frac{\sin A - \sin C}{\cos C - \cos A} = \cot B \] This suggests a relationship between the angles \( A \), \( B \), and \( C \) in the triangle.
Step 2: Verifying the progression.
By solving the equation and comparing the relationship between the angles, we find that \( A \), \( B \), and \( C \) are in arithmetic progression.
Step 3: Conclusion.
Thus, the angles \( A \), \( B \), and \( C \) are in arithmetic progression, which makes option (D) the correct answer.