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 

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

Answer 2

To evaluate the trigonometric expression \(4 \cos \frac{\pi}{7} \cos \frac{\pi}{5} \cos \frac{2\pi}{7} \cos \frac{2\pi}{5} \cos \frac{4\pi}{7}\), we can utilize properties of cosine angles and product-to-sum identities. First, note that the cosines involved are periodic and symmetric: \(\cos \frac{4\pi}{7}=-\cos \frac{3\pi}{7}\). This property helps simplify calculations. Another useful identity for evaluating such products is the multiple angle formula: \(\cos nx = T_n(\cos x)\), where \(T_n\) is the Chebyshev polynomial. For the angles given, express them using \(e^{ix}\) to take advantage of complex roots, where \(z^7=1\) (roots of unity). Specifically, extracting terms related to cosine: consider \(z=e^{i\pi/7}\), and note key properties: \[ \cos \frac{\pi}{7} = \frac{1}{2}(z + z^{-1}) \] \[ \cos \frac{2\pi}{7} = \frac{1}{2}(z^2 + z^{-2}) \] \[ \cos \frac{4\pi}{7} = \frac{1}{2}(z^4 + z^{-4}) \] Consequently, \( \cos \frac{\pi}{5} \) and \( \cos \frac{2\pi}{5} \) can be deduced from \( \cos \frac{\pi}{5}=\frac{\sqrt{5}+1}{4} \) and \(\cos \frac{2\pi}{5}=\frac{\sqrt{5}-1}{4}\). Let's implement these transformations: \[ \left(\cos \frac{\pi}{5} \cos \frac{2\pi}{5}\right) = \frac{\sqrt{5}^2-1^2}{16}=\frac{1}{4} \] Substitute back into the main expression, compute: \[ 4 \cos \frac{\pi}{7} \cos \frac{2\pi}{7} \cos \frac{4\pi}{7} \times \frac{1}{4} \] Expands to: \[ 4 \cdot \left(\frac{1}{4}\right) = 1 \] Therefore, coordinate transformations yield: \[ -\frac{1}{8} \] as the expression value. Thus, the correct option is \(-\frac{1}{8}\).