### Trigonometry: Compound Angles

Package of problems

## Problem

Throughout, $A$, $B$ and $C$ are the angles of a triangle.

For each of the following, decide whether it is an identity (true for all triangles) or an equation (there is a triangle for which it is not true).

If it is an identity, true for all triangles, then you should prove it (using trigonometric identities that you already know).

If it is an equation, then at the very least you should give an example of a triangle for which it is not true. You could also try to solve the equation (that is, find all triangles for which it is true).

1. $\sin(A + 2B) = \sin A + 2\sin B \cos(A + B)$.

2. $\tan(A - B) + \tan(B - C) + \tan(C - A) = 0$.

3. $2\sin A \cos^2\left(\frac{B}{2}\right) + 2\cos^2\left(\frac{A}{2}\right)\sin B = \sin(A + B) + \sin(B + C) + \sin(C + A)$.

4. $\sin(A+B) = \cos C$.

5. $\cos C = -\cos(A+B)$.

6. $4(\cos^2 A \cos^2 B + \sin^2 A \sin^2 B) - 2 \sin(2A) \sin(2B) = 3$.

7. $\sin(2A) + \sin(2B) + \sin(2C) = 4\sin A \sin B \sin C$.