Rewrite tan^4x in terms of cosine of multiple angles with no power greater than 1

Answer 1

will be using cos2x=cos²x-sin²x

and since cos²x+sin²x=1

2sin²x=1-cos2x

2cos²x=1+cos2x

tan⁴x=(1-cos2x)²/(1+cos2x)²

=(1-2cos2x+cos²2x)/(1+cos2x+cos²2x)

cos²2x=(1/2)(1+cos4x)

tan⁴x=((3/2-2cos2x+(cos4x)/2)/((3/2)+2cos2x+(cos4x)/2)

=(3-4cos2x+cos4x)/(3+4cos2x+cos4x)