<p>If <span>\(\cos\theta - \sin\theta = \cos\alpha - \sin\alpha\)</span>, then the value of <span>\(|\theta + \alpha|\)</span> is:</p>
Step-by-Step Solution
Key Concept: Rewrite the equation as cos θ - sin θ = cos α - sin α, then recognize that the function f(x) = cos x - sin x can be expressed as √2 cos(x + π/4), which is monotonic in restricted intervals. For the equation f(θ) = f(α) to hold with distinct angles, use the property that cos θ - sin θ = cos α - sin α implies either θ = α or θ + α = nπ.
<p><strong>Step 1:</strong> Express cos θ - sin θ in a simpler form using the formula a cos x + b sin x = √(a² + b²) cos(x + φ):</p><p>cos θ - sin θ = √2 cos(θ + π/4)</p><p><strong>Step 2:</strong> Similarly, cos α - sin α = √2 cos(α + π/4)</p><p><strong>Step 3:</strong> Given condition becomes: √2 cos(θ + π/4) = √2 cos(α + π/4)</p><p>This gives us: cos(θ + π/4) = cos(α + π/4)</p><p><strong>Step 4:</strong> For cos A = cos B, we have A = ±B + 2πn</p><p>Case 1: θ + π/4 = α + π/4 + 2πn ⟹ θ = α (gives |θ + α| = 2|θ|, not fixed)</p><p>Case 2: θ + π/4 = -(α + π/4) + 2πn ⟹ θ + α = -π/2 + 2πn</p><p><strong>Step 5:</strong> For the principal value (taking n = 0): θ + α = -π/2</p><p>Therefore: |θ + α| = π/2</p><p>∴ Answer: B</p>
Correct Answer: B