Finding the derivative of sinus and cosinus. Trigonometric identities How can we see that $$\sin(x+h)-\sin(x)=2\sin\left(\frac h2\right)\cos\left(x+\frac h2\right)$$
How can we see that $$\cos(x+h)-\cos(x)=-2\sin\left(\frac h2\right)\sin\left(x+\frac h2\right)$$
Do these identities have a name?

For any $a,b$, we have the well known "addition theorems"
$$ \sin(a+b) = \sin a \cos b + \sin b \cos a$$
and $$ \sin(a-b) = \sin a \cos b - \sin b \cos a $$
Subtracting these two equations, we get 
$$ \sin(a+b) - \sin(a-b) = 2\sin b \cos a $$
For the cosine, we have
$$ \cos(a+b) = \cos a \cos b - \sin a \sin b $$
and $$ \cos(a-b) = \cos a \cos b + \sin a \sin b $$
Subtraction again 
$$ \cos(a+b) - \cos(a-b) =  -2\sin a\sin b$$
Now let $b = \frac h2$, $a = x+\frac h2$.
