Question

Evaluate the given trigonometric expression: \[ 4 \cos \frac{\pi}{7} \cos \frac{\pi}{5} \cos \frac{2\pi}{7} \cos \frac{2\pi}{5} \cos \frac{4\pi}{7} = \]

• A. \( -\frac{1}{8} \)
• B. \( \frac{1}{32} \)
• C. \( -\frac{1}{32} \)
• D. \( \frac{1}{8} \)

Hint:

When evaluating products of multiple cosine terms, it is useful to recognize standard trigonometric identities for products of cosine terms with symmetric angles.

Answer 1

The Correct Option is A

We are tasked with evaluating the given trigonometric expression: \[ 4 \cos \frac{\pi}{7} \cos \frac{\pi}{5} \cos \frac{2\pi}{7} \cos \frac{2\pi}{5} \cos \frac{4\pi}{7} \] Step 1: Group the Terms We'll begin by grouping the terms in pairs to simplify the product. \[ = 4 \left(\cos \frac{\pi}{7} \cos \frac{2\pi}{7} \cos \frac{4\pi}{7} \right) \left( \cos \frac{\pi}{5} \cos \frac{2\pi}{5} \right) \] Step 2: Evaluate Each Group Part 1: \( \cos \frac{\pi}{7} \cos \frac{2\pi}{7} \cos \frac{4\pi}{7} \) From the identity: \[ \cos \frac{\pi}{7} \cos \frac{2\pi}{7} \cos \frac{4\pi}{7} = -\frac{1}{8} \] Part 2: \( \cos \frac{\pi}{5} \cos \frac{2\pi}{5} \) From the identity: \[ \cos \frac{\pi}{5} \cos \frac{2\pi}{5} = \frac{1}{4} \] Step 3: Combine Results \[ 4 \left(-\frac{1}{8} \right) \left(\frac{1}{4} \right) \] \[ = 4 \times -\frac{1}{32} \] \[ = -\frac{1}{8} \] Step 4: Final Answer 

\[Correct Answer: (1) \ -\frac{1}{8}\]