<p>The equation \(\dfrac{1 - \cos x}{\cos x} = (\sqrt{2} - 1)\dfrac{\sin x}{\cos x}\) simplifies to find solutions in \([0, 2\pi]\). Which of the following are solutions?</p>
Step-by-Step Solution
Key Concept: Divide both sides by cos(x) (valid when cos x ≠ 0), then rearrange to get a tangent-based equation. Recognize that 1 - cos(x) = 2sin²(x/2) and sin(x) = 2sin(x/2)cos(x/2) enables factoring, or directly work with the algebraic form to isolate tan(x).
<p><strong>Step 1:</strong> Start with the equation: <br/>$$\frac{1 - \cos x}{\cos x} = (\sqrt{2} - 1)\frac{\sin x}{\cos x}$$</p><p><strong>Step 2:</strong> Multiply both sides by cos(x) (valid when cos(x) ≠ 0):<br/>$$1 - \cos x = (\sqrt{2} - 1)\sin x$$</p><p><strong>Step 3:</strong> Rearrange:<br/>$$1 = \cos x + (\sqrt{2} - 1)\sin x$$</p><p><strong>Step 4:</strong> This is of the form 1 = cos(x) + k·sin(x). Let's rewrite by dividing by a suitable factor or testing key angles. Express in the form: cos(x) + (\sqrt{2} - 1)sin(x) = 1.</p><p><strong>Step 5:</strong> Test critical angles in [0, 2π]. For x = π/4: cos(π/4) + (√2 - 1)sin(π/4) = (√2/2) + (√2 - 1)(√2/2) = (√2/2)[1 + √2 - 1] = (√2/2)(√2) = 1 ✓</p><p><strong>Step 6:</strong> For x = 0: cos(0) + (√2 - 1)sin(0) = 1 + 0 = 1 ✓</p><p><strong>Step 7:</strong> For x = 2π: cos(2π) + (√2 - 1)sin(2π) = 1 + 0 = 1 ✓</p><p><strong>Step 8:</strong> Verify no solution exists at cos(x) = 0 (x = π/2, 3π/2) since original equation requires cos(x) ≠ 0.</p><p>∴ Answer: <strong>A, B, C</strong> (corresponding to x = 0, x = π/4, and x = 2π, depending on option labeling)</p>
Correct Answer: A,B,C