Trigonometry & Inverse Trigonometry
Trigonometric identities
Grade 11
Question:
<p>If \(f_4(x) - f_6(x) = \frac{1}{4}(\sin^4 x + \cos^4 x) - \frac{1}{6}(\cos^6 x + \sin^6 x)\), then the value of this expression equals:</p>
<p>\(\frac{1}{12}\)</p>
<p>\(\frac{1}{6}\)</p>
<p>\(\frac{1}{4}\)</p>
<p>\(\frac{1}{3}\)</p>
Step-by-Step Solution
Key Concept: Reduce the expression using algebraic identities: express sin⁴x + cos⁴x and sin⁶x + cos⁶x in terms of sin²x·cos²x, then simplify using the constraint sin²x + cos²x = 1.
<p><strong>Step 1:</strong> Express sin⁴x + cos⁴x using the identity:</p><p>sin⁴x + cos⁴x = (sin²x + cos²x)² - 2sin²x·cos²x = 1 - 2sin²x·cos²x</p><p><strong>Step 2:</strong> Express sin⁶x + cos⁶x using the factorization a³ + b³:</p><p>sin⁶x + cos⁶x = (sin²x)³ + (cos²x)³ = (sin²x + cos²x)[(sin²x + cos²x)² - 3sin²x·cos²x]</p><p>= 1·[1 - 3sin²x·cos²x] = 1 - 3sin²x·cos²x</p><p><strong>Step 3:</strong> Substitute into the original expression:</p><p>f₄(x) - f₆(x) = (1/4)(1 - 2sin²x·cos²x) - (1/6)(1 - 3sin²x·cos²x)</p><p>= 1/4 - (1/2)sin²x·cos²x - 1/6 + (1/2)sin²x·cos²x</p><p>= 1/4 - 1/6 = (3 - 2)/12 = 1/12</p><p><strong>Step 4:</strong> The expression is independent of x and equals a constant.</p><p>∴ Answer: A (which corresponds to 1/12)</p>
Correct Answer: A