Trigonometry

Let fk(x) = 1/k(sink x + cosk 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.