12sin2xcos2x

The expression 12sin2xcos2x is a trigonometric expression involving the sine and cosine functions, each applied to the argument 2x, and then multiplied by a constant factor of 12. This expression can be simplified using trigonometric identities.

One relevant identity is the double angle formula for sine, which states that sin(2θ) = 2sin(θ)cos(θ). While the expression involves 2x as the argument, we can consider θ = 2x. In this case, the identity would be sin(2(2x)) = 2sin(2x)cos(2x), which simplifies to sin(4x) = 2sin(2x)cos(2x).

Looking at the original expression, 12sin2xcos2x, we can see that it contains the term sin2xcos2x. By comparing this to the double angle identity, we can rewrite sin2xcos2x as (1/2)sin(2 * 2x), which is (1/2)sin(4x).

Therefore, substituting this back into the original expression, we get 12 * [(1/2)sin(4x)]. This simplifies to 6sin(4x).

Thus, the expression 12sin2xcos2x is equivalent to 6sin(4x). This simplification is useful in various contexts, such as solving trigonometric equations, performing integration, or analyzing the behavior of functions involving these terms. The transformation allows for a more compact representation and can make subsequent mathematical operations easier.