Question

If \[ \frac{\cos^2 48^\circ - \sin^2 12^\circ}{\sin^2 24^\circ - \sin^2 6^\circ} = \frac{\alpha + \beta\sqrt{5}}{2}, \] where \( \alpha, \beta \in \mathbb{N} \), then the value of \( \alpha + \beta \) is ___________.

Hint:

Expressions involving squares of trigonometric functions are best simplified using double-angle identities.

Answer 1

Correct Answer: 6

Step 1: Use trigonometric identities.
\[ \cos^2 A - \sin^2 B = \frac{1}{2}(\cos 2A + \cos 2B), \quad \sin^2 C - \sin^2 D = \frac{1}{2}(\cos 2D - \cos 2C). \] Step 2: Apply the identities.
\[ \cos^2 48^\circ - \sin^2 12^\circ = \tfrac{1}{2}(\cos 96^\circ + \cos 24^\circ), \] \[ \sin^2 24^\circ - \sin^2 6^\circ = \tfrac{1}{2}(\cos 12^\circ - \cos 48^\circ). \] Step 3: Simplify using standard angle values.
After simplification, the expression becomes \[ \frac{6 + \sqrt{5}}{2}. \] Hence,
\[ \alpha = 6, \quad \beta = 1. \] Final Answer:
\[ \boxed{6} \]