Question

If \( \alpha, \beta, \gamma \in [0, \pi] \) and if \( \alpha, \beta, \gamma \) are in AP, then \[ \frac{\sin \alpha - \sin \gamma}{\cos \gamma - \cos \alpha} \] {is equal to:}

• A. \( \sin \beta \)
• B. \( \cos \beta \)
• C. \( \cot \beta \)
• D. \( 2 \cos \beta \)

Hint:

When solving problems involving sequences in AP, express the terms in terms of the middle term (here, \( \beta \)), and use standard trigonometric identities such as sum-to-product formulas to simplify the expression.

Answer 1

The Correct Option is C

We start with the given identities and simplify them as follows: \[ \frac{\sin \alpha - \sin \gamma}{\cos \gamma - \cos \alpha} = \frac{2\cos\left(\frac{\alpha + \gamma}{2}\right) \sin\left(\frac{\alpha - \gamma}{2}\right)}{-2\sin\left(\frac{\alpha + \gamma}{2}\right) \sin\left(\frac{\alpha - \gamma}{2}\right)} \] \[ = \cot \left(\frac{\alpha + \gamma}{2}\right) + \frac{1}{2} \] This is based on the identities: \[ \sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \] \[ \cos A - \cos B = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \] If $\alpha, \beta, \gamma$ are in arithmetic progression, then: \[ \frac{\alpha + \gamma}{2} = \beta \] Thus, the required value is: \[ \cot \beta \] Therefore, the correct answer is Option C.