<p>The number of solutions of the equation \(|\cot x| = \cot x + \frac{1}{\sin x}\), \(0 < x < 2\pi\) is</p>
Step-by-Step Solution
Key Concept: The absolute value equation |cot x| = cot x + 1/sin x must be split into cases based on the sign of cot x. Case 1 (cot x ≥ 0) gives cot x = cot x + 1/sin x, which leads to sin x = -∞ (impossible). Case 2 (cot x < 0) gives -cot x = cot x + 1/sin x, which simplifies to a solvable trigonometric equation.
<p><strong>Step 1: Set up case analysis for |cot x|</strong></p><p>The equation is |cot x| = cot x + 1/sin x, where 0 < x < π.</p><p>We must consider two cases:</p><p><strong>Case 1: cot x ≥ 0</strong> (which means cos x · sin x ≥ 0)</p><p>Then |cot x| = cot x, so: cot x = cot x + 1/sin x</p><p>This gives: 0 = 1/sin x, or sin x = ∞, which is impossible.</p><p>No solutions from Case 1.</p><p><strong>Step 2: Analyze Case 2 where cot x < 0</strong></p><p>Then |cot x| = -cot x, so: -cot x = cot x + 1/sin x</p><p>This gives: -2cot x = 1/sin x</p><p>Rewrite as: -2(cos x/sin x) = 1/sin x</p><p>Multiply by sin x (valid since sin x ≠ 0 for 0 < x < π):</p><p>-2cos x = 1</p><p>Therefore: cos x = -1/2</p><p><strong>Step 3: Solve cos x = -1/2 in the interval (0, π)</strong></p><p>In (0, π), cos x = -1/2 gives x = 2π/3.</p><p><strong>Step 4: Verify the solution satisfies cot x < 0</strong></p><p>At x = 2π/3: sin(2π/3) = √3/2 > 0 and cos(2π/3) = -1/2 < 0</p><p>Therefore cot(2π/3) = cos(2π/3)/sin(2π/3) = (-1/2)/(√3/2) = -1/√3 < 0 ✓</p><p>The condition for Case 2 is satisfied.</p><p><strong>Step 5: Verify the solution in the original equation</strong></p><p>At x = 2π/3:</p><p>LHS: |cot(2π/3)| = |-1/√3| = 1/√3</p><p>RHS: -1/√3 + 1/(√3/2) = -1/√3 + 2/√3 = 1/√3 ✓</p><p><strong>∴ Answer:</strong> A (There is exactly 1 solution)</p>
Correct Answer: A