Trigonometry & Inverse Trigonometry
Product and Sum Formulas
Grade 11
Question:
<p>The value of \(\cos\frac{\pi}{15}\cos\frac{2\pi}{15}\cos\frac{4\pi}{15}\cos\frac{8\pi}{15}\) is</p>
<p>(a) \(\frac{1}{16}\)</p>
<p>(b) \(-\frac{1}{16}\)</p>
<p>(c) 1</p>
<p>(d) 0</p>
Step-by-Step Solution
Key Concept: Use product-to-sum formulas to convert products of cosines into sums, then simplify using known cosine values.
<p><strong>Step 1:</strong> Apply product-to-sum formula strategically to simplify the product.</p><p><strong>Step 2:</strong> $\cos\frac{\pi}{15}\cos\frac{2\pi}{15}\cos\frac{4\pi}{15}\cos\frac{8\pi}{15}$</p><p><strong>Step 3:</strong> Group terms: $\left(\cos\frac{\pi}{15}\cos\frac{8\pi}{15}\right)\left(\cos\frac{2\pi}{15}\cos\frac{4\pi}{15}\right)$</p><p><strong>Step 4:</strong> Using $\cos A\cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)]$, simplify to get $-\frac{1}{16}$</p><p>∴ Answer is (b) $-\frac{1}{16}$</p>
Correct Answer: B