Answer: I don't know if you wanted to write sin(4x) or [tex]sin^{4} (x)[/tex] , but here we go:
ok, sin(4x) = sin(2x + 2x), and we know that:
sin (a + b) = sin(a)*cos(b) + sin(b)*cos(a)
then sin (2x + 2x) = sin(2x)*cos(2x) + cos(2x)*sin(2x) = 2cos(2x)*sin(2x)
So 2*cos(2x)*sin(2x) is equivalent of sin(4x)
If you writed [tex]sin^{4} (x)[/tex] then:
[tex]sin^{4} (x) = sin^{2} (x)*sin^{2} (x)[/tex]
and using that: [tex]sins^{2} (x) = \frac{1-cos(2x)}{2}[/tex]
we have: [tex]sin^{2} (x)*sin^{2} (x) = \frac{(1-cos(2x))*(1-cos(2x))}{4} = \frac{1-2cos(2x) + cos^{2}(2x) }{4}[/tex]
and using that: [tex]cos^{2} (x) = \frac{1 + cos(2x)}{2}[/tex]
[tex]\frac{1-2cos(2x) + cos^{2}(2x) }{4} = \frac{1-2cos(2x) + \frac{1+cos(4x)}{2} }{4}[/tex]
You can keep simplifying it, but there is your representation of [tex]sin^{4} (x)[/tex] that does not contain powers of trigonometric functions greater than 1.
