If cos 2 48 - sin 2 12 sin 2 24 - sin 2 6 = 5 2 , where , then is equal to __________ . [2026]

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Solution

Q.
If $\frac{\cos^{2} 48 ° - \sin^{2} 12 °}{\sin^{2} 24 ° - \sin^{2} 6 °} = \frac{α + β \sqrt{5}}{2},$ where $α, β \in ℕ$ , then $α + β$ is equal to __________ . [2026]

Ans.
(4)

$Use \sin (A + B) \sin (A - B) = \sin^{2} A - \sin^{2} B$

$\cos (A + B) \cos (A - B) = \cos^{2} A - \sin^{2} B$

$\frac{\cos 60 ° \cos 36 °}{\sin 30 ° \sin 18 °} = \frac{\sqrt{5} + 1}{\sqrt{5} - 1} \times \frac{\sqrt{5} + 1}{\sqrt{5} + 1} = \frac{{(\sqrt{5} + 1)}^{2}}{4}$

$= \frac{3 + \sqrt{5}}{2}$

$α = 3, β = 1$

$So, (α + β) = 4$

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