<p>The number of solutions of \(\sin x \cdot \tan 4x = \cos x\) in \(\left(0, \pi\right)\) is:</p>
Step-by-Step Solution
Key Concept: Rewrite the equation as sin(x)·tan(4x) = cos(x) → tan(4x) = cot(x), then use tan(4x) = cot(x) = tan(π/2 - x) to find when 4x = π/2 - x + nπ, giving x = π/10 + nπ/5. Finally, check which solutions lie in (0,π) and don't make tan(4x) undefined.
<p><strong>Step 1:</strong> Start with sin(x)·tan(4x) = cos(x). For cos(x) ≠ 0, divide both sides by cos(x):</p><p>tan(x)·tan(4x) = 1, or equivalently tan(4x) = cot(x) = tan(π/2 - x)</p><p><strong>Step 2:</strong> The general solution is 4x = π/2 - x + nπ, which gives:</p><p>5x = π/2 + nπ → x = π/10 + nπ/5</p><p><strong>Step 3:</strong> Find all values in (0, π):</p><p>• n = 0: x = π/10 ✓</p><p>• n = 1: x = π/10 + π/5 = 3π/10 ✓</p><p>• n = 2: x = π/10 + 2π/5 = 5π/10 = π/2 ✓</p><p>• n = 3: x = π/10 + 3π/5 = 7π/10 ✓</p><p>• n = 4: x = π/10 + 4π/5 = 9π/10 ✓</p><p>• n = 5: x = π/10 + π = 11π/10 > π ✗</p><p><strong>Step 4:</strong> Verify none of these make tan(4x) undefined. Check 4x = π/2, 3π/2, 5π/2,... None of our x-values satisfy this.</p><p><strong>Step 5:</strong> Verify tan(x)·tan(4x) = 1 for each solution (all satisfy the original equation).</p><p>∴ Answer: <strong>C</strong> (5 solutions)
Correct Answer: C