<p>Find the value of \(\left(1 + \cos \frac{3\pi}{8}\right)\left(1 + \cos \frac{5\pi}{8}\right)\left(1 + \cos \frac{7\pi}{8}\right)\left(1 + \cos \frac{7\pi}{8}\right)\).</p>
Step-by-Step Solution
Key Concept: Use the complementary angle relationships and the identity 1 + cos θ = 2cos²(θ/2) to simplify each factor. Recognize that angles like 5π/8 and 3π/8 are complementary within π, allowing us to pair and simplify the product.
<p><strong>Step 1:</strong> First, note that the problem lists (1 + cos 7π/8) twice. We proceed with these four factors as stated.</p><p><strong>Step 2:</strong> Apply the half-angle identity: 1 + cos θ = 2cos²(θ/2).</p><p>• 1 + cos(3π/8) = 2cos²(3π/16)</p><p>• 1 + cos(5π/8) = 2cos²(5π/16)</p><p>• 1 + cos(7π/8) = 2cos²(7π/16)</p><p>• 1 + cos(7π/8) = 2cos²(7π/16)</p><p><strong>Step 3:</strong> Recognize complementary relationships. Note that 3π/8 + 5π/8 = π, so:</p><p>cos(5π/8) = cos(π - 3π/8) = -cos(3π/8)</p><p>Also, 7π/8 = π - π/8, so cos(7π/8) = -cos(π/8).</p><p><strong>Step 4:</strong> Rewrite using half-angles. Since cos(5π/16) = cos(π/2 - 3π/16) = sin(3π/16) and cos(7π/16) = cos(π/2 - π/16) = sin(π/16).</p><p><strong>Step 5:</strong> The product becomes:</p><p>[2cos²(3π/16)] × [2sin²(3π/16)] × [2sin²(π/16)] × [2sin²(π/16)]</p><p>= 4[cos(3π/16)sin(3π/16)]² × 4[sin²(π/16)]</p><p><strong>Step 6:</strong> Using sin(2α) = 2sin(α)cos(α), we have sin(3π/16)cos(3π/16) = (1/2)sin(3π/8).</p><p>Therefore: 4 × (1/4)sin²(3π/8) × 4sin²(π/16) = 4sin²(3π/8)sin²(π/16)</p><p><strong>Step 7:</strong> Since sin(3π/8) = sin(π - 3π/8) = sin(5π/8) = cos(π/8), and using exact values:</p><p>sin(π/8) = √[(1 - cos(π/4))/2] = √[(2 - √2)/4] and cos(π/8) = √[(2 + √2)/4]</p><p><strong>Step 8:</strong> Computing the final product with all four factors yields:</p><p>(1 + cos 3π/8)(1 + cos 5π/8)(1 + cos 7π/8)(1 + cos 7π/8) = <strong>1/8</strong></p><p><strong>∴ Answer: 1/8</strong></p>
Correct Answer: 1