2sinucosu

The expression "2sin(u)cos(u)" is a trigonometric identity commonly encountered in mathematics, particularly in calculus and trigonometry. It represents a product of the sine and cosine functions of the same angle, scaled by a factor of two. This expression is notable for its relationship to the double-angle identity for sine, which simplifies certain trigonometric computations.

The identity can be derived using Euler's formula or by applying the sine addition formula. Specifically, the sine of a sum formula states that sin(a + b) = sin(a)cos(b) + cos(a)sin(b). When a = b = u, this becomes sin(2u) = 2sin(u)cos(u). This demonstrates that 2sin(u)cos(u) is equivalent to sin(2u), a key simplification in trigonometric proofs and calculations.

In practical applications, this identity is useful in integrating trigonometric functions, solving differential equations, and analyzing periodic phenomena such as waves and oscillations. For example, when integrating sin²(u) or cos²(u), substitution techniques often rely on this identity to reduce the complexity of the integral.

The expression is also fundamental in physics, particularly in the study of harmonic motion, where trigonometric identities help model periodic behavior. Additionally, it appears in signal processing, where it aids in the analysis of waveforms and frequency components.

While 2sin(u)cos(u) is most commonly associated with the double-angle identity, it can also be generalized to other forms, such as 2sin(au)cos(bu), which may appear in more advanced trigonometric manipulations. Understanding this identity is essential for students and professionals working in fields that involve trigonometric analysis.