What is the formula for the sine double-angle identity?

The sine double-angle identity states that the sine of a double angle (2θ) can be expressed as twice the product of the sine and cosine of the angle (θ). This relationship is represented by the formula sin(2θ) = 2sin(θ)cos(θ).

Understanding this identity is crucial because it simplifies calculations involving trigonometric functions, particularly in problems that require the evaluation of sine at angles that are multiples of a given angle. The identity allows one to rewrite sin(2θ) in terms of sin(θ) and cos(θ), thus enabling easier computation and manipulation of trigonometric expressions.

In the context of the other options, the definitions do not accurately convey the sine double-angle identity. For example, the first option incorrectly combines sine and cosine squares in a manner that does not derive the sine of a double angle. The third option suggests a relationship involving sine squared alone, which is not representative of the sine double-angle identity. The fourth option inaccurately combines sine and cosine linearly and does not reflect the necessary multiplicative relationship between sin(θ) and cos(θ) that characterizes the double angle.

Therefore, option B clearly represents the correct sine double-angle identity, illustrating the direct