<p>If \(\cos^2 x = t\), and the equation \(5\left[\frac{1-t}{t} - t\right] = 2(2t-1) + 9\) is satisfied, then find the value of \(\cos 4x\).</p>
Step-by-Step Solution
Key Concept: Substitute cos²x = t to get a quadratic in t, solve for t, then use the double angle formula cos 2x = 2cos²x - 1 twice to find cos 4x = 2cos²2x - 1.
<p><strong>Step 1: Clear the denominator and simplify</strong></p><p>Given: 5[(1-t)/t - t] = 2(2t-1) + 9, where t = cos²x</p><p>Multiply by t: 5[(1-t) - t²] = t[4t - 2 + 9]</p><p>5[1 - t - t²] = t(4t + 7)</p><p>5 - 5t - 5t² = 4t² + 7t</p><p><strong>Step 2: Form and solve quadratic</strong></p><p>9t² + 12t - 5 = 0</p><p>Using quadratic formula: t = [-12 ± √(144 + 180)]/18 = [-12 ± 18]/18</p><p>t = 1/3 or t = -5/3</p><p>Since t = cos²x ∈ [0,1], we have t = 1/3</p><p><strong>Step 3: Find cos 2x</strong></p><p>cos 2x = 2cos²x - 1 = 2(1/3) - 1 = -1/3</p><p><strong>Step 4: Find cos 4x</strong></p><p>cos 4x = 2cos²2x - 1 = 2(1/9) - 1 = 2/9 - 1 = -7/9</p><p>∴ Answer: cos 4x = <strong>-7/9</strong></p>
Correct Answer: C