2sin2xcos2x

The expression 2sin2xcos2x is a trigonometric identity that can be simplified. It is derived from the double angle formula for sine, which states that sin(2A) = 2sinAcosA. In this case, if we let A = 2x, then the expression becomes sin(2 * 2x).

Applying the double angle formula, we substitute 2x for A:

sin(2 * 2x) = 2sin(2x)cos(2x).

Therefore, the expression 2sin2xcos2x is equivalent to sin(4x). This simplification is useful in various areas of trigonometry, including calculus and the solving of trigonometric equations.

The identity sin(4x) = 2sin(2x)cos(2x) demonstrates a recursive application of the double angle formula. It shows how a sine function with an argument of 4x can be expressed in terms of trigonometric functions of an argument of 2x. This type of simplification can be helpful when evaluating integrals or derivatives of trigonometric functions, or when seeking to find the roots of more complex trigonometric equations. The transformation from 2sin2xcos2x to sin(4x) reduces the number of trigonometric functions and their arguments, often making the expression easier to manipulate.