Trigonometry & Inverse Trigonometry
Trigonometric Identities
Grade 11
Question:
<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>
<p>1</p>
<p>2</p>
<p>\(\dfrac{1}{2}\)</p>
<p>\(\dfrac{-1}{2}\)</p>
Step-by-Step Solution
Key Concept: From cos x + cos²x = 1, derive that sin²x = cos x by using sin²x + cos²x = 1. Then express the given expression E entirely in terms of sin²x to find its numerical value.
<p><strong>Step 1:</strong> From the constraint cos x + cos²x = 1, use sin²x + cos²x = 1 to get:</p><p>sin²x = 1 - cos²x = cos x</p><p><strong>Step 2:</strong> Let u = sin²x = cos x. Then:</p><p>E = sin¹²x + 3sin¹⁰x + 3sin⁸x + sin⁶x + 2</p><p>E = u⁶ + 3u⁵ + 3u⁴ + u³ + 2</p><p><strong>Step 3:</strong> Recognize that u⁶ + 3u⁵ + 3u⁴ + u³ = u³(u³ + 3u² + 3u + 1) = u³(u + 1)³</p><p>So E = u³(u + 1)³ + 2 = [u(u + 1)]³ + 2</p><p><strong>Step 4:</strong> Since u = cos x and u + 1 = 1 + cos x = 1 + sin²x, from cos x + cos²x = 1 we get cos x(1 + cos x) = cos²x + cos x = 1</p><p>Therefore: [u(u + 1)]³ = 1³ = 1</p><p><strong>Step 5:</strong> Thus E = 1 + 2 = 3</p><p><strong>Step 6:</strong> log₍ₜₐₙ(π/3)₎ E = log₍√₃₎ 3 = log₍√₃₎ (√3)² = 2</p><p>∴ Answer: <strong>A (= 2)</strong></p>
Correct Answer: A