The particular identity you want to use is
[tex]\cos^2x=\dfrac{1+\cos(2x)}2[/tex]
Then
[tex]3\cos^4x=3(\cos^2x)^2=3\left(\dfrac{1+\cos(2x)}2\right)^2=\dfrac34(1+\cos(2x))^2[/tex]
Expand the binomial to get
[tex]3\cos^4x=\dfrac34\left(1+2\cos(2x)+\cos^2(2x)\right)[/tex]
Use the identity again to write
[tex]\cos^2(2x)=\dfrac{1+\cos(4x)}2[/tex]
and so
[tex]3\cos^4x=\dfrac34\left(1+2\cos(2x)+\dfrac{1+\cos(4x)}2\right)[/tex]
[tex]3\cos^4x=\dfrac38\left(3+4\cos(2x)+\cos(4x)\right)[/tex]
