2002 AMC 12P Problem 10

Let f_n (x) = \text{sin}^n x + \text{cos}^n x. For how many x in [0,\pi] is it true that

6f_{4}(x)-4f_{6}(x)=2f_{2}(x)?

\text{(A) }2 \qquad \text{(B) }4 \qquad \text{(C) }6 \qquad \text{(D) }8 \qquad \text{(E) more than }8

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.