Question

Let \( s_n = \cos\left(\frac{n\pi}{10}\right) \), \( n=1,2,3, \ldots \). Then the value of \( \frac{s_1s_2\cdots s_{10}}{s_1+s_2+\cdots+s_{10}} \) is equal to:

• A. \( \frac{1}{\sqrt{2}} \)
• B. \( \frac{\sqrt{3}}{2} \)
• C. \( 2\sqrt{2} \)
• D. \( 0 \)
• E. \( \frac{1}{2} \)

Hint:

Whenever you see a long product of trigonometric terms, check for "special angles" like $90^\circ$ for cosine or $0^\circ/180^\circ$ for sine that might zero out the whole expression.

Answer 1

The Correct Option is D

Concept: Check the individual terms of the product in the numerator. If any single term is zero, the entire product is zero.

Step 1: Evaluate the terms in the numerator.
The numerator is the product \( s_1 \cdot s_2 \cdot s_3 \cdot \ldots \cdot s_{10} \). Specifically, consider \( s_5 \): \[ s_5 = \cos\left(\frac{5\pi}{10}\right) = \cos\left(\frac{\pi}{2}\right) \]

Step 2: Determine the value of the zero term.
We know that \( \cos\left(\frac{\pi}{2}\right) = 0 \).

Step 3: Calculate the final ratio.
Since \( s_5 = 0 \), the product \( s_1s_2\cdots s_{10} = 0 \). Assuming the sum in the denominator is non-zero: \[ \frac{0}{\text{Sum}} = 0 \]