What are double angle identities used for in trigonometry?

Double angle identities are fundamental formulas in trigonometry that allow mathematicians and students to express trigonometric functions of multiple angles in terms of functions of single angles. These identities involve the sine, cosine, and tangent functions, and provide a way to simplify expressions involving double angles—specifically, angles that are twice a given angle.

For instance, the double angle formulas include expressions like ( \sin(2\theta) = 2\sin(\theta)\cos(\theta) ) and ( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) ). These identities enable the simplification of complex trigonometric equations and are essential in various applications such as solving problems in geometry, physics, and engineering.

The other options focus on different areas of mathematics that do not directly relate to the specific purpose of double angle identities. Determining the circumference of circles pertains to geometry and the relationships of circles rather than trigonometric identities. Calculating angles in non-right triangles typically involves the Law of Sines or the Law of Cosines. Finally, finding the derivatives of trigonometric functions falls within calculus, not trigonometric identities themselves. Thus, understanding double angle identities is