Please help me to solve this, use the power-reducing formulas to rewrite the expression in terms of first powers of the cosines of multiple angles.
#sin^4(5x) cos^2(5x)#?

Question

Zoe Austin · Accepted Answer

#sin^4(5x) cos^2(5x)# #=1/2(2sin^2(5x))*1/4(2sin(5x) cos(5x))^2# #=1/8(2sin^2(5x))(2sin(5x) cos(5x))^2# #=1/8(1-cos(10x))*sin^2(10x)# #=1/16(1-cos(10x))*2sin^2(10x)# #=1/16(1-cos(10x))*(1-cos(20x))#
#=1/16(1-cos(10x)-cos(20x)+cos(10x)cos(20x))# #=1/16(1-cos(10x)-cos(20x)+1/2(cos(15x)+cos(5x)))# #=1/32(2-2cos(10x)-2cos(20x)+cos(15x)+cos(5x))#

Madelyn Adams · Answer

undefined To rewrite the expression in terms of first powers of the cosines of multiple angles, we can use the power-reducing formula for sine and cosine:

sin^2(x) = (1 - cos(2x))/2
cos^2(x) = (1 + cos(2x))/2

Substitute these formulas into the expression:

sin^4(5x) cos^2(5x) = (sin^2(5x))^2 cos^2(5x)
                      = ((1 - cos(10x))/2)^2 ((1 + cos(10x))/2)
                      = ((1 - cos(10x))^2 / 4) * ((1 + cos(10x)) / 2)
                      = ((1 - 2cos(10x) + cos^2(10x)) / 4) * ((1 + cos(10x)) / 2)
                      = (1 - 2cos(10x) + cos^2(10x)) * (1 + cos(10x)) / 8
                      = (1 + cos(10x) - 2cos(10x) - 2cos^2(10x)) / 8
                      = (1 - cos(10x) - 2cos^2(10x)) / 8

So, sin^4(5x) cos^2(5x) rewritten in terms of first powers of the cosines of multiple angles is:

(1 - cos(10x) - 2cos^2(10x)) / 8

Please help me to solve this, use the power-reducing formulas to rewrite the expression in terms of first powers of the cosines of multiple angles. #sin^4(5x) cos^2(5x)#?