Answer:
[tex]4(\sin(2\theta))^2[/tex]
Step-by-step explanation:
We can simplify the trigonometric expression:
[tex]4 (\cos(\theta))^2 \cdot 4(\sin(\theta))^2[/tex]
with the following steps:
1) grouping like terms
[tex]=4^2(\cos(\theta))^2 (\sin(\theta))^2[/tex]
2) rewriting each factor squared as the square of the product
[tex]=\left(\dfrac{}{}4\cos(\theta)\sin(\theta)\dfrac{}{}\right)^\!2[/tex]
3) replacing the expression in parentheses with the double angle identity:
[tex]\sin(2\theta) = 2\sin(\theta)\cos(\theta)[/tex]
↓↓↓
[tex]=\left(\dfrac{}{}2\cdot \underline{2\cos(\theta)\sin(\theta)}\dfrac{}{}\right)^\!2[/tex]
[tex]=\left(\dfrac{}{}2\sin(2\theta)\dfrac{}{}\right)^2[/tex]
4) distributing the square to each factor
[tex]= \boxed{4(\sin(2\theta))^2}[/tex]
