Final answer:
The question involves verifying trigonometric identities, including the double-angle identities for sine and cosine, as well as applying the Law of Sines and the Law of Cosines.
Step-by-step explanation:
The question pertains to verifying trigonometric identities, which is a common type of problem in high school mathematics. Various trigonometric equations are provided, with instructions to prove or verify them. For example, one such equation is sin(2θ) = 2sin(θ)cos(θ), which is known as the double-angle identity for sine. Another is cos(2θ) = cos2(θ) - sin2(θ), which can be further transformed into 2cos2(θ) - 1 or 1 - 2sin2(θ), showcasing the versatility of trigonometric identities.
Additionally, we see references to the Law of Sines and the Law of Cosines, which are critical in solving triangle problems involving side lengths and angle measures. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides and angles. The Law of Cosines is a generalized form of the Pythagorean theorem and helps determine a side length when two sides and the included angle are known.