Answer:
We want to rewrite:
sin^4(x)
in a way that only involves the first power of the cosine function.
We know that:
sin^2(x) = 1 - cos^2(x)
Then:
sin^4(x) = sin^2(x)*sin^2(x) = sin^2(x)*(1 - cos^2(x))
Now we know that:
cos(2x) = cos^2(x) - sin^2(x)
then
cos(2x) + sin^2(x) = cos^2(x)
We can replace that in our equation to get:
sin^2(x)*(1 - cos^2(x)) = sin^2(x)*(1 - cos(2x) - sin^2(x) )
sin^2(x)*(1 - cos(2x) - sin^2(x) ) = sin^2(x) - sin^2(x)*cos(2x) - sin^4(x)
then:
sin^4(x) = sin^2(x) - sin^2(x)*cos(2x) - sin^4(x)
sin^4(x) + sin^4(x) = sin^2(x) - sin^2(x)*cos(2x)
2*sin^4(x) = sin^2(x) - sin^2(x)*cos(2x) = sin^2(x)*(1 - cos(2x))
sin^4(x) = sin^2(x)*(1 - cos(2x))/2
So we rewrite the equation in such a way that we only have the first power of the cosine function.
