Question

Rewrite sin^4x that involves only the first power of cos​

Answer 1

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.