Question

The value of \( \cos^{2} 48^\circ - \sin^{2} 12^\circ \) is:

• A. $\frac{\sqrt{5}-1}{4}$
• B. $\frac{\sqrt{5}+1}{8}$
• C. $\frac{\sqrt{3}-1}{4}$
• D. $\frac{\sqrt{5}+1}{5}$

Hint:

Memorizing values like $\sin 18^{\circ} = \frac{\sqrt{5}-1}{4}$ and $\cos 36^{\circ} = \frac{\sqrt{5}+1}{4}$ is very helpful for competitive exams.

Answer 1

The Correct Option is B

Step 1: Concept
Use the identity $\cos^{2} A - \sin^{2} B = \cos(A+B)\cos(A-B)$.
Step 2: Analysis
Let $A = 48^{\circ}$ and $B = 12^{\circ}$.
Result $= \cos(48^{\circ} + 12^{\circ})\cos(48^{\circ} - 12^{\circ}) = \cos 60^{\circ} \cos 36^{\circ}$.
We know $\cos 60^{\circ} = 1/2$ and $\cos 36^{\circ} = \frac{\sqrt{5}+1}{4}$.
Step 3: Conclusion
Result $= \frac{1}{2} \times \frac{\sqrt{5}+1}{4} = \frac{\sqrt{5}+1}{8}$.
Final Answer: (B)