$$ \sin(a)\cos(b) = \frac{1}{2}(\sin(a-b) + \sin(a+b)) $$
$$ \sin(a)\sin(b) = \frac{1}{2}(\cos(a-b) - \cos(a+b)) $$
$$ \cos(a)\cos(b) = \frac{1}{2}(\cos(a-b) + \cos(a+b)) $$
How do you derive these formulas from other Trig Identities such as any of:
- Angle Addition Identities
- Pythagorean Trig Identity
- Half Angle Identities
- Double Angle Identities
Or, if it can't be proven from other identities, what's the simplest way to prove them? Would prefer not to have a geometric proof from scratch for these, and be able to remember them by just remembering other more basic ones.
Thanks to commenters for explaining this:
$$\begin{align} \sin(a-b)&=\sin(a)\cos(b)−\sin(b)\cos(a) \\ \\ \sin(a+b)&=\sin(a)\cos(b)+\sin(b)\cos(a) \\ \\ \sin(a-b) + \sin(a+b) &= 2\sin(a)\cos(b) \\ \\ \sin(a)\cos(b) &= \frac{\sin(a-b) + \sin(a+b)}{2} \end{align}$$