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

For each of the following, decide whether it is an

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