Question:
Evaluate
\[
\cos\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{3\pi}{7}
\cos\frac{4\pi}{7}\cos\frac{5\pi}{7}\cos\frac{6\pi}{7}.
\]
Evaluate
\[
\cos\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{3\pi}{7}
\cos\frac{4\pi}{7}\cos\frac{5\pi}{7}\cos\frac{6\pi}{7}.
\]
Show Hint
Remember the standard identity
\[
\cos\frac{\pi}{7}
\cos\frac{2\pi}{7}
\cos\frac{3\pi}{7}
=
\frac18.
\]
For products involving all six cosine terms, pair them using
\[
\cos(\pi-\theta)=-\cos\theta
\]
to reduce the expression quickly.
Updated On: Jul 9, 2026
- \(\dfrac{1}{8}\)
- \(-\dfrac{1}{16}\)
- \(\dfrac{1}{32}\)
- \(-\dfrac{1}{64}\) \bigskip
Show Solution
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The Correct Option is D
Solution and Explanation
Concept:
Use the identity
\[
\cos(\pi-\theta)=-\cos\theta.
\]
This allows us to pair terms and reduce the product to a known standard result:
\[
\cos\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{3\pi}{7}
=\frac{1}{8}.
\]
Step 1: Pair the terms using \(\cos(\pi-\theta)=-\cos\theta\). Observe that \[ \cos\frac{4\pi}{7} = -\cos\frac{3\pi}{7}, \] \[ \cos\frac{5\pi}{7} = -\cos\frac{2\pi}{7}, \] \[ \cos\frac{6\pi}{7} = -\cos\frac{\pi}{7}. \] Therefore, \[ P= \cos\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{3\pi}{7} \cos\frac{4\pi}{7}\cos\frac{5\pi}{7}\cos\frac{6\pi}{7} \] \[ = -\left( \cos\frac{\pi}{7} \cos\frac{2\pi}{7} \cos\frac{3\pi}{7} \right)^2. \]
Step 2: Use the standard trigonometric product. The well-known identity is \[ \cos\frac{\pi}{7} \cos\frac{2\pi}{7} \cos\frac{3\pi}{7} = \frac{1}{8}. \] Substituting, \[ P = -\left(\frac18\right)^2. \] \[ P = -\frac{1}{64}. \]
Step 3: Write the final answer. \[ \boxed{-\frac{1}{64}} \]
Step 1: Pair the terms using \(\cos(\pi-\theta)=-\cos\theta\). Observe that \[ \cos\frac{4\pi}{7} = -\cos\frac{3\pi}{7}, \] \[ \cos\frac{5\pi}{7} = -\cos\frac{2\pi}{7}, \] \[ \cos\frac{6\pi}{7} = -\cos\frac{\pi}{7}. \] Therefore, \[ P= \cos\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{3\pi}{7} \cos\frac{4\pi}{7}\cos\frac{5\pi}{7}\cos\frac{6\pi}{7} \] \[ = -\left( \cos\frac{\pi}{7} \cos\frac{2\pi}{7} \cos\frac{3\pi}{7} \right)^2. \]
Step 2: Use the standard trigonometric product. The well-known identity is \[ \cos\frac{\pi}{7} \cos\frac{2\pi}{7} \cos\frac{3\pi}{7} = \frac{1}{8}. \] Substituting, \[ P = -\left(\frac18\right)^2. \] \[ P = -\frac{1}{64}. \]
Step 3: Write the final answer. \[ \boxed{-\frac{1}{64}} \]
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