Question

$\cos 75^{\circ} \cos 45^{\circ} \cos 15^{\circ} = $ ________.

• A. $\frac{1}{3\sqrt{2}}$
• B. $\frac{1}{\sqrt{2}}$
• C. $\frac{1}{4\sqrt{2}}$
• D. $\frac{1}{2\sqrt{3}}$
• E. $\frac{2}{\sqrt{3}}$

Hint:

$2\sin\theta\cos\theta = \sin(2\theta)$.

Answer 1

The Correct Option is C

Step 1: Concept
Use the product-to-sum formula or substitute known values.

Step 2: Meaning
$\cos 75^{\circ} = \sin 15^{\circ}$ and $\cos 45^{\circ} = \frac{1}{\sqrt{2}}$.

Step 3: Analysis
$(\sin 15^{\circ} \cos 15^{\circ}) \cos 45^{\circ} = \frac{1}{2}(2 \sin 15^{\circ} \cos 15^{\circ}) \frac{1}{\sqrt{2}} = \frac{1}{2}(\sin 30^{\circ}) \frac{1}{\sqrt{2}}$.

Step 4: Conclusion
$\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{\sqrt{2}} = \frac{1}{4\sqrt{2}}$. Final Answer: (C)