We have to reduce the expression:
[tex]4\sin^4(2x)[/tex]With power-reducing formulas we use identities that let us replace higher exponents terms with lower exponents terms.
We can start using the following identity:
[tex]\sin^2\theta=\frac{1}{2}(1-\cos2\theta)[/tex]Replacing in our formula we obtain:
[tex]\begin{gathered} 4\sin^4(2x)=4[\frac{1}{2}(1-\cos(4x))]^2 \\ 4\sin^4(2x)=4\cdot(\frac{1}{2})^2[1-\cos(4x)]^2 \\ 4\sin^4(2x)=4\cdot\frac{1}{4}\cdot(1^2-2\cos(4x)+\cos^2(4x)) \\ 4\sin^4(2x)=1-2\cos(4x)+\cos^2(4x) \end{gathered}[/tex]We can replace the last term using this identity:
[tex]\cos^2\theta=\frac{1}{2}(1+\cos2\theta)[/tex]Then, we will obtain:
[tex]\cos^2(4x)=\frac{1}{2}[1+\cos(8x)][/tex]Replacing in the equation we obtain:
[tex]\begin{gathered} 1-2\cos(4x)+\cos^2(4x) \\ 1-2\cos(4x)+\frac{1}{2}(1+\cos(8x)) \\ 1-2\cos(4x)+\frac{1}{2}+\frac{1}{2}\cos(8x) \\ \frac{3}{2}-2\cos(4x)+\frac{1}{2}\cos(8x) \end{gathered}[/tex]Answer:
3/2 - 2*cos(4x) + 1/2*cos(8x)