Question

Simplify the following expression: \[ \cos 18^\circ \cos 42^\circ \cos 78^\circ. \] The simplified form is:

• A. \( \frac{1}{4} \cos 36^\circ \)
• B. \( \frac{1}{4} \cos 72^\circ \)
• C. \( \frac{1}{\sin 72^\circ} \)
• D. \( \frac{1}{4} \sin 36^\circ \)
• E. None of the above

Hint:

When simplifying products of trigonometric functions, use product-to-sum identities to reduce terms, and use known trigonometric values to simplify the result.

Answer 1

The Correct Option is D

We are asked to simplify the expression \( \cos{18^\circ} \cos{42^\circ} \cos{78^\circ} \). To solve this, we can use trigonometric identities and relationships between the angles. Start by noting that: \[ \cos{18^\circ} \cos{42^\circ} \cos{78^\circ} = \frac{1}{4} \sin{36^\circ} \] This result comes from the fact that the product of cosines of angles which sum up to specific angles (like the sum of \(18^\circ + 42^\circ + 78^\circ\)) can be simplified using known identities or patterns that relate the angles to sine and cosine values.
Thus, the correct answer is option (D), \( \frac{1}{4} \sin{36^\circ} \).