Question

If \( \cos\alpha + \cos\beta = \frac{1}{3} \) and \( \sin\alpha + \sin\beta = \frac{1}{4} \), then \( \cos(\alpha + \beta) = \)

• A. \( \frac{24}{25} \)
• B. \( \frac{7}{25} \)
• C. \( \frac{13}{25} \)
• D. \( \frac{12}{13} \)

Hint:

For identities involving sums, square both sides and use algebraic identities to isolate the desired trigonometric product.

Answer 1

The Correct Option is B

Step 1: Use identity: \[ \cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta \]
Step 2: Use sum identities: \[ (\cos\alpha + \cos\beta)^2 = \cos^2\alpha + \cos^2\beta + 2\cos\alpha\cos\beta = \frac{1}{9} \] \[ (\sin\alpha + \sin\beta)^2 = \sin^2\alpha + \sin^2\beta + 2\sin\alpha\sin\beta = \frac{1}{16} \]
Step 3: Subtract and simplify: \[ \cos\alpha\cos\beta = \frac{1}{2} \left[ \frac{1}{9} - \cos^2\alpha - \cos^2\beta \right], \text{ similarly for sine terms} \] Final: \[ \cos(\alpha + \beta) = \boxed{\frac{7}{25}} \]