Question

If \( \sin x + \sin y = \alpha \), \( \cos x + \cos y = \beta \), then \( \csc(x + y) \) =

• A. \( \frac{\beta^2 - \alpha^2}{\beta^2 + \alpha^2} \)
• B. \( \frac{2 \alpha \beta}{\beta^2 - \alpha^2} \)
• C. \( \frac{\beta^2 + \alpha^2}{2 \alpha \beta} \)
• D. \( \frac{2 \alpha \beta}{\beta^2 + \alpha^2} \)

Hint:

Use the sum-to-product identities to simplify the given trigonometric expressions and solve for the required function.

Answer 1

The Correct Option is C

We are given the following equations: \[ \sin x + \sin y = \alpha, \quad \cos x + \cos y = \beta. \] We need to find the value of \( \csc(x + y) \). 
Step 1: We will use the sum-to-product identities to express \( \sin x + \sin y \) and \( \cos x + \cos y \) in a different form. The sum-to-product identities are: \[ \sin x + \sin y = 2 \sin \left( \frac{x + y}{2} \right) \cos \left( \frac{x - y}{2} \right), \] \[ \cos x + \cos y = 2 \cos \left( \frac{x + y}{2} \right) \cos \left( \frac{x - y}{2} \right). \] Substitute the given values \( \sin x + \sin y = \alpha \) and \( \cos x + \cos y = \beta \): \[ \alpha = 2 \sin \left( \frac{x + y}{2} \right) \cos \left( \frac{x - y}{2} \right), \] \[ \beta = 2 \cos \left( \frac{x + y}{2} \right) \cos \left( \frac{x - y}{2} \right). \] 
Step 2: Divide the equation for \( \alpha \) by the equation for \( \beta \): \[ \frac{\alpha}{\beta} = \frac{2 \sin \left( \frac{x + y}{2} \right) \cos \left( \frac{x - y}{2} \right)}{2 \cos \left( \frac{x + y}{2} \right) \cos \left( \frac{x - y}{2} \right)}. \] Simplifying this: \[ \frac{\alpha}{\beta} = \tan \left( \frac{x + y}{2} \right). \] 
Step 3: We know that: \[ \csc(x + y) = \frac{1}{\sin(x + y)}. \] We can express \( \sin(x + y) \) as: \[ \sin(x + y) = 2 \sin \left( \frac{x + y}{2} \right) \cos \left( \frac{x + y}{2} \right). \] From the previous step, we have \( \tan \left( \frac{x + y}{2} \right) = \frac{\alpha}{\beta} \), so: \[ \sin \left( \frac{x + y}{2} \right) = \frac{\alpha}{2 \cos \left( \frac{x + y}{2} \right)}. \] Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we get: \[ \sin^2 \left( \frac{x + y}{2} \right) = \frac{\alpha^2}{4 \beta^2}, \] \[ \cos^2 \left( \frac{x + y}{2} \right) = \frac{\beta^2}{4 \beta^2}. \] 
Step 4: Now, substituting the values into the equation for \( \csc(x + y) \), we get: \[ \csc(x + y) = \frac{\beta^2 + \alpha^2}{2 \alpha \beta}. \] Thus, the correct answer is \( \frac{\beta^2 + \alpha^2}{2 \alpha \beta} \).