The angle addition formula says that:
$\sin(\phi + \theta) = \sin(\phi) \cdot \cos(\theta) + \cos(\phi) \cdot \sin(\theta)$
Why are the following steps valid?:
$\sin(\phi − \theta) = \sin(\phi) \cdot \cos(−\theta) + \cos(\phi) \cdot \sin(−\theta)= \sin(\phi) \cdot \cos(\theta) − \cos(\phi) \cdot \sin(\theta)$
Thanks.
Hint:
$\cos(\theta)$ is an even function while $\sin(\theta)$ is an odd function.
Hence:
$$\cos(-\theta) = \cos(\theta),$$
while
$$\sin(-\theta) = -\sin(\theta).$$
(See this Wikipedia link on Even and odd functions for more information.)
They're valid because of the following identities:
$\qquad\sin(-\theta) \equiv -\sin(\theta) \qquad\text{since sin is an}$ odd function.
$\qquad \cos(-\theta) \equiv \cos(\theta) \qquad\text{since cos is an} $ even function.
The $\equiv$ symbol means "is equal to for all (permitted) values of $\ \theta$".
They are valid because $\sin(-x) = -\sin(x) \forall_{x \in [0, 2\pi]}$ (sinus is an odd function) and $\cos(-x) = \cos(x) \forall_{x \in [0, 2\pi]}$ (cosine is an even function). You can see this by looking at sinus and cosine graphics.
