Question

\[ \cos 6^\circ \sin 24^\circ \cos 72^\circ= \]

• A. \(\frac{1}{4}\)
• B. \(-\frac{1}{8}\)
• C. \(-\frac{1}{4}\)
• D. \(\frac{1}{8}\)

Hint:

For products of trigonometric functions, use product-to-sum identities to simplify quickly.

Answer 1

The Correct Option is D

Concept: Use product-to-sum identities and known trigonometric angle relations.

Step 1: Given: \[ \cos 6^\circ \sin 24^\circ \cos 72^\circ \]

Step 2: Use product-to-sum identity: \[ 2\sin A\cos B=\sin(A+B)+\sin(A-B) \] First combine: \[ 2\sin 24^\circ \cos 72^\circ = \sin 96^\circ+\sin(-48^\circ) \] \[ =\sin 96^\circ-\sin 48^\circ \]

Step 3: Now multiply with \(\cos6^\circ\).
Using standard trigonometric simplification, the complete product reduces to: \[ \frac{1}{8} \] Therefore, \[ \boxed{\frac{1}{8}} \]