Use the powerreducing formulas to rewrite the given expressi

Use the power-reducing formulas to rewrite the given expression in terms of the first power of the cosine.

Solution

sin⁴x

=(sin²x)²

=(1-cos²x)²(since sin²a+cos²a=1)

=1-2cos²x+cos⁴x

=1-2cos²x+(cos²x)²

=1-(2(1+cos2x)/2)+((1+cos2x)/2)²  (since cos²a =(1+cos2a)/2)

=1-1-cos2x +((1+cos2x)/2)²

= -cos2x +(1/4)(1+cos2x)²

= -cos2x +(1/4)(1+2cos2x+(cos2x)²)

= -cos2x +(1/4)+(1/2)cos2x+(1/4)(cos2x)²

= -cos2x +(1/4)+(1/2)cos2x+(1/4)((1+cos4x)/2)

=(1/4)-(1/2)cos2x+ (1/8)(1+cos4x)

=(1/4)-(1/2)cos2x+ (1/8)+ (1/8)cos4x

=(3/8)-(1/2)cos2x+ (1/8)cos4x

sin⁴x=(3/8)-(1/2)cos2x+ (1/8)cos4x