<p><strong>173.</strong> If \(\cos x + \cos^2 x = 1\). Let \(E = \sin^{12} x + 3\sin^{10} x + 3\sin^8 x + \sin^6 x + 2\), then the value of \(\log_{\tan\frac{\pi}{3}} E\) is:</p>
Step-by-Step Solution
Key Concept: From cos x + cos²x = 1, derive that sin²x = cos x, then systematically express all powers of sin x in terms of cos x to simplify the polynomial expression E.
<p><strong>Step 1:</strong> From the constraint cos x + cos²x = 1, rearrange to get cos²x = 1 - cos x.</p><p>Since sin²x + cos²x = 1, we have sin²x = 1 - cos²x = 1 - (1 - cos x) = cos x.</p><p><strong>Step 2:</strong> Using sin²x = cos x, express all even powers of sin x:</p><p>• sin⁶x = (sin²x)³ = cos³x</p><p>• sin⁸x = (sin²x)⁴ = cos⁴x</p><p>• sin¹⁰x = (sin²x)⁵ = cos⁵x</p><p>• sin¹²x = (sin²x)⁶ = cos⁶x</p><p><strong>Step 3:</strong> Substitute into E:</p><p>E = cos⁶x + 3cos⁵x + 3cos⁴x + cos³x + 2</p><p><strong>Step 4:</strong> Recognize the pattern. Note that (cos x + 1)³ = cos³x + 3cos²x + 3cos x + 1.</p><p>From cos x + cos²x = 1, we get cos²x = 1 - cos x, so cos x + 1 = cos²x + cos x + 1 = (1 - cos x) + cos x + 1 = 2.</p><p>Therefore: E = cos⁶x + 3cos⁵x + 3cos⁴x + cos³x + 2 = (cos²x + cos x)³ + 2 = 1³ + 2 = 3</p><p><strong>Step 5:</strong> Calculate log_{tan(π/3)} E = log₍√₃₎ 3 = log₍√₃₎ (√3)² = 2</p><p>∴ Answer: A (value is 2)</p>
Correct Answer: A