Question

\[ \sin9^\circ\sin18^\circ\sin36^\circ\sin54^\circ\sin72^\circ\sin81^\circ \]

• A. \(\frac{10+2\sqrt5}{128}\)
• B. \(\frac{5-\sqrt5}{128}\)
• C. \(\frac{5-\sqrt5}{64}\)
• D. \(\frac{10-2\sqrt5}{512}\) \bigskip

Hint:

Use angle pairing \( \theta + (90^\circ-\theta)=90^\circ \).

Answer 1

The Correct Option is B

Concept: Use complementary angle pairing: \[ \sin\theta\sin(90^\circ-\theta) \]

Step 1: Pair terms.
\[ \sin9^\circ\sin81^\circ=\frac12\sin18^\circ \] \[ \sin18^\circ\sin72^\circ=\frac12\sin36^\circ \] \[ \sin36^\circ\sin54^\circ=\frac12\sin72^\circ \]

Step 2: Multiply.
\[ = \frac{1}{8}\sin18^\circ\sin36^\circ\sin72^\circ \] Using known identity: \[ \sin18^\circ\sin36^\circ\sin72^\circ=\frac{5-\sqrt5}{16} \]

Step 3: Final result.
\[ =\frac{5-\sqrt5}{128} \] \[ \boxed{(B)} \]