<p>If 2cos θ + 2√2 = 3 sec θ, where θ ∈ (0, 2π), then which of the following can be correct?</p>
Step-by-Step Solution
Key Concept: Convert the trigonometric equation to a quadratic in cos θ, solve for cos θ, then determine the corresponding angles and verify all possible trigonometric values.
<p><strong>Step 1:</strong> Given equation: 2cos θ + 2√2 = 3 sec θ = 3/cos θ</p><p><strong>Step 2:</strong> Multiply by cos θ: 2cos² θ + 2√2 cos θ - 3 = 0</p><p><strong>Step 3:</strong> Using quadratic formula:</p><p>cos θ = (-2√2 ± √(8 + 24))/4 = (-2√2 ± 4√2)/4</p><p><strong>Step 4:</strong> cos θ = √2/2 = 1/√2 or cos θ = -3√2/2 (rejected as |cos θ| ≤ 1)</p><p><strong>Step 5:</strong> cos θ = 1/√2 ⟹ θ = π/4 or θ = 7π/4</p><p><strong>Step 6:</strong> At θ = π/4: sin θ = 1/√2, tan θ = 1, cot θ = 1</p><p>At θ = 7π/4: sin θ = -1/√2, tan θ = -1, cot θ = -1</p><p>∴ All options (a), (b), (c), (d) can be correct.</p>
Correct Answer: A,B,C,D