Use the formula for lowering powers to rewrite the expressio

Use the formula for lowering powers to rewrite the expression given below as an equivalent expression containing terms of the first power of cosine (All factors need to be expanded and your final answer should contain terms separated with plus and minus signs): 6 sin^4 x

Solution

sin²x=(1_/2)(1-cos(2x)),cos²x=(1_/2)(1+cos(2x))

6sin⁴x

=6(sin²x)²

=6((1_/2)(1-cos(2x)))²

=6(1_/4)(1-cos(2x))²

=(3_/2)(1²-2cos(2x)+cos²(2x))

=(3_/2)(1-2cos(2x)+cos²(2x))

=(3_/2)(1-2cos(2x)+ (1_/2)(1+cos(2*2x)))

=(3_/2)(1-2cos(2x)+ (1_/2)+(1_/2)cos(4x))

=(3_/2)((3_/2)-2cos(2x)+ (1_/2)cos(4x))

=(3_/2)²-2*(3_/2)cos(2x)+ (3_/2)(1_/2)cos(4x)

=(9_/4)-3cos(2x)+ (3_/4)cos(4x)