Question

If \[ \cos \alpha = k \cos \beta \] then \[ \cot \left( \frac{\alpha + \beta}{2} \right) \] is equal to

• A. \( \frac{k + 1}{k - 1} \tan \left( \frac{\alpha - \beta}{2} \right) \)
• B. \( \frac{k + 1}{k - 1} \tan \left( \frac{\beta - \alpha}{2} \right) \)
• C. \( \frac{k + 1}{k + 1} \tan \left( \frac{\alpha - \beta}{2} \right) \)
• D. \( \frac{k - 1}{k + 1} \tan \left( \frac{\alpha - \beta}{2} \right) \)

Hint:

When working with cotangent and trigonometric identities, remember the angle sum and difference formulas and how they can simplify your calculations.

Answer 1

The Correct Option is B

We are given that \( \cos \alpha = k \cos \beta \), and we are asked to find an expression for \( \cot \left( \frac{\alpha + \beta}{2} \right) \). Step 1: Use sum and difference trigonometric identities First, recall the trigonometric identity for the cotangent of a sum: \[ \cot \left( \frac{\alpha + \beta}{2} \right) = \frac{\cos \left( \frac{\alpha + \beta}{2} \right)}{\sin \left( \frac{\alpha + \beta}{2} \right)} \] We can relate this to the given equation \( \cos \alpha = k \cos \beta \) using angle sum identities. Step 2: Manipulate the equation and simplify Using the relationship between \( \cos \alpha \) and \( \cos \beta \), and simplifying using the angle sum identities, we find that: \[ \cot \left( \frac{\alpha + \beta}{2} \right) = \frac{k + 1}{k - 1} \tan \left( \frac{\beta - \alpha}{2} \right) \] Thus, the correct answer is \( \frac{k + 1}{k - 1} \tan \left( \frac{\beta - \alpha}{2} \right) \).