Let fk(x) = 1/k(sin^k x + cos^k x) where x ∈ R and k ≥ 1

• Trigonometry

Let f_k(x) = 1/k(sin^k x + cos^k x) where x ∈ R and k ≥ 1. Then f₄(x) - f₆(x) equals

1. 1/12
2. 1/6
3. 1/4
4. 1/3

Answer

1/4(sin⁴ x - cos⁴ x) - 1/6(sin⁶ x - cos⁶ x)

= [3(sin⁴ x - cos⁴ x) - 2(sin⁶ x - cos⁶ x)] / 12

= [3(1 - 2sin² x cos² x) - 2(1 - 3sin² x cos² x)] / 12

The correct option is A.