# Expand #sin^6 x#?

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We want to expand

#sin^6(x)#

One way is to use these identities repeatedly

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

This often gets quite long, which sometimes leads to mistake

Another way is to use the complex numbers (and Euler's formula)

We can express sine and cosine as

#color(red)(sin(x)=(e^(ix)-e^(-ix))/(2i))# and#color(red)(cos(x)=(e^(ix)+e^(-ix))/2)#

Thus

The third way is using De Moivre's theorem, we can express

#color(blue)(2cos(nx)=z^n+1/z^n)# and#color(blue)(2isin(nx)=z^n-1/z^n#

where

Thus

#(2isin(x))^6=(z-1/z)^6#

#=>sin^6(x)=-1/64(z-1/z)^6#

Expand the binomial on the right hand side

Thus

#sin^6(x)=-1/64(2cos(6x)-12cos(4x)+30cos(2x)-20)#

#sin^6(x)=-1/32(cos(6x)-6cos(4x)+15cos(2x)-10)#

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