<p>If \(\cos\theta + \sec\theta = 2\) then \(\cos^n\theta + \sec^n\theta\) is equal to</p>
Step-by-Step Solution
Key Concept: From cos θ + sec θ = 2, deduce that cos θ = 1, then use this constraint to evaluate cos^n θ + sec^n θ for any positive integer n.
<p><strong>Step 1:</strong> Use the constraint cos θ + sec θ = 2.</p><p>Since sec θ = 1/cos θ, we have: cos θ + 1/cos θ = 2</p><p><strong>Step 2:</strong> Multiply by cos θ: cos²θ + 1 = 2cos θ</p><p>Rearrange: cos²θ - 2cos θ + 1 = 0</p><p>(cos θ - 1)² = 0</p><p><strong>Step 3:</strong> This gives cos θ = 1, which means sec θ = 1 as well.</p><p><strong>Step 4:</strong> Therefore: cos^n θ + sec^n θ = 1^n + 1^n = 1 + 1 = 2</p><p>∴ Answer: <strong>2</strong> (for any positive integer n)</p>
Correct Answer: B