Final answer:
The question involves trigonometric identities and laws, such as the Pythagorean identity and equations for the sum of sines and cosines. These are used to express one trigonometric function in terms of another, specifically in the context of the relationships within triangles as defined by the law of sines and the law of cosines.
Step-by-step explanation:
The student seems to be working with trigonometric identities and laws that relate various trigonometric functions to each other. For instance, converting expressions in terms of sin and cos could involve using the Pythagorean identity cos2(θ) + sin2(θ) = 1. The expression given in point 14 of the question, cos 2θ = cos2(θ) - sin2(θ), can also be written in terms of sin only by using this Pythagorean identity to get 1 - 2 sin2(θ).
Similarly, trigonometric expressions for the sum of sines or cosines can be found in identities number 15 and 16: sin a + sin β = 2 sin((a + β)/2) cos((a - β)/2), and cos a + cos β = 2 cos((a + β)/2) cos((a - β)/2). In triangles, the law of sines and the law of cosines relate the lengths of sides to the sines and cosines of the angles.