<p>If \(\cos(\theta - \alpha)\), \(\cos\theta\), \(\cos(\theta + \alpha)\) are in HP, then \(\cos\theta \sec\dfrac{\alpha}{2}\) is equal to</p>
Step-by-Step Solution
Key Concept: If three terms are in HP, their reciprocals are in AP. Use this to convert the HP condition into an AP condition for secants, then apply sum-to-product formulas strategically.
<p><strong>Step 1:</strong> If cos(θ - α), cos θ, cos(θ + α) are in HP, then their reciprocals are in AP.</p><p>∴ sec(θ - α), sec θ, sec(θ + α) are in AP</p><p><strong>Step 2:</strong> AP condition gives: 2 sec θ = sec(θ - α) + sec(θ + α)</p><p><strong>Step 3:</strong> Rewrite: 2/cos θ = 1/cos(θ - α) + 1/cos(θ + α)</p><p><strong>Step 4:</strong> Finding common denominator on RHS:</p><p>2/cos θ = [cos(θ + α) + cos(θ - α)]/[cos(θ - α)cos(θ + α)]</p><p><strong>Step 5:</strong> Apply sum-to-product: cos(θ + α) + cos(θ - α) = 2 cos θ cos α</p><p>2/cos θ = 2 cos θ cos α/[cos(θ - α)cos(θ + α)]</p><p><strong>Step 6:</strong> Simplify: 1/cos θ = cos θ cos α/[cos(θ - α)cos(θ + α)]</p><p>∴ [cos(θ - α)cos(θ + α)]/cos²θ = cos α</p><p><strong>Step 7:</strong> Using product formula: cos(θ - α)cos(θ + α) = cos²θ - sin²α</p><p>(cos²θ - sin²α)/cos²θ = cos α</p><p>1 - sin²α/cos²θ = cos α</p><p><strong>Step 8:</strong> sin²α = (1 - cos α)cos²θ = 2sin²(α/2)cos²θ</p><p>Using sin²α = 4sin²(α/2)cos²(α/2):</p><p>cos θ sec(α/2) = <strong>±√2</strong></p><p>∴ Answer: <strong>C</strong> (±√2)</p>
Correct Answer: C