<p>The number of solutions of \(|\cos x| = \sin x\) such that \(0 < x < \pi\) is</p>
Step-by-Step Solution
Key Concept: We must consider two cases based on the sign of cos x, since |cos x| = cos x when cos x ≥ 0 and |cos x| = -cos x when cos x < 0. Each case yields a different trigonometric equation to solve.
<p><strong>Step 1:</strong> Split into cases based on the sign of cos x.</p><p><strong>Case 1: cos x ≥ 0</strong></p><p>Then |cos x| = cos x, so the equation becomes:</p><p>cos x = sin x</p><p>This gives tan x = 1, so x = π/4 or x = π/4 + π = 5π/4</p><p>Check: At x = π/4, cos(π/4) = 1/√2 > 0 ✓ and sin(π/4) = 1/√2 ✓</p><p>Check: At x = 5π/4, cos(5π/4) = -1/√2 < 0 ✗ (violates case condition)</p><p></p><p><strong>Case 2: cos x < 0</strong></p><p>Then |cos x| = -cos x, so the equation becomes:</p><p>-cos x = sin x</p><p>This gives sin x + cos x = 0, or tan x = -1, so x = 3π/4 or x = 7π/4</p><p>Check: At x = 3π/4, cos(3π/4) = -1/√2 < 0 ✓ and -cos(3π/4) = 1/√2 = sin(3π/4) ✓</p><p>Check: At x = 7π/4, cos(7π/4) = 1/√2 > 0 ✗ (violates case condition)</p><p></p><p><strong>Step 2:</strong> Verify the solutions in (0, 2π).</p><p>Valid solutions: x = π/4 and x = 3π/4</p><p></p><p><strong>Step 3:</strong> Count the number of solutions.</p><p>There are exactly 2 solutions in the interval (0, 2π).</p><p>∴ Answer: B</p>
Correct Answer: B