<p>If \(\sin^4\alpha + 4\cos^4\beta + 2 = 4\sqrt{2}\sin\alpha\cos\beta\); \(\alpha, \beta \in [0, \pi]\), then \(\cos(\alpha+\beta) - \cos(\alpha-\beta) =\) __________ (up to four decimal places).</p>
Step-by-Step Solution
Key Concept: Recognize this as a hidden sum-of-squares constraint by rearranging the equation into the form (a - b)² + (c - d)² = 0, which forces individual terms to equal zero simultaneously. This determines the unique values of sin α and cos β.
<p><strong>Step 1:</strong> Rearrange the equation sin⁴α + 4cos⁴β + 2 = 4√2 sin α cos β as a sum of squares.</p><p>sin⁴α - 2√2 sin α + 2 + 4cos⁴β - 2√2 sin α + 0 = 0</p><p>Rewrite as: (sin²α - √2)² + (2cos²β - √2)² = 0</p><p><strong>Step 2:</strong> Since a sum of squares equals zero, each term must be zero:</p><p>sin²α - √2 = 0 ⟹ sin²α = √2, but sin α ∈ [0,1], so sin α = 1</p><p>2cos²β - √2 = 0 ⟹ cos²β = √2/2 ⟹ cos β = ±1/√2</p><p><strong>Step 3:</strong> Determine α and β from the constraints α, β ∈ [0, π]:</p><p>sin α = 1 ⟹ α = π/2</p><p>cos β = 1/√2 (taking positive root for β ∈ [0, π]) ⟹ β = π/4</p><p><strong>Step 4:</strong> Calculate cos(α + β) - cos(α - β):</p><p>cos(π/2 + π/4) - cos(π/2 - π/4) = cos(3π/4) - cos(π/4)</p><p>= -1/√2 - 1/√2 = -2/√2 = -√2</p><p>∴ Answer: <strong>-1.4142</strong> or <strong>-√2</strong></p>
Correct Answer: -\sqrt{2}