1. The problem is to simplify or solve the expression $1 - 2 \sin^2(2x)$. 2. Recall the double-angle identity for cosine: $$\cos(2\theta) = 1 - 2\sin^2(\theta)$$. This means that $$1 - 2\sin^2(2x) = \cos(4x)$$ because here $\theta = 2x$. 3. So, the expression simplifies directly to $$\cos(4x)$$. 4. This is a useful simplification because it converts a sine squared term into a cosine term with a double angle, which can be easier to work with in equations or integrals. 5. Therefore, $$1 - 2\sin^2(2x) = \cos(4x)$$ is the simplified form of your left side.