Different ways finding the derivative of $\sin$ and $\cos$. I am looking for different ways of differentiating $\sin$ and $\cos$, especially when using the geometric definition, but ways that use other defintions are also welcome. 
Please include the definition you are using. 
I think there are several ways to do this I didn't find, that make clever use of trigonometric identities or derivative facts. I am looking for these for several reasons. Firstly, it is just interesting to see. Secondly, there may be interesting techniques that can also be applied in a clever way to other derivative problems. It is also interesting to see how proofs can come form completely different fields of mathematics or even from physics.
I have included several solutions in the answers. 

To the first order, the tangent is a good approximation of the boundary of a shape.
So can you approximate $(\cos(\theta+d\theta),\sin(\theta+d\theta))$ in terms of $(\cos\theta,\sin\theta)$? In other words, what's:
$(\cos(\theta+d\theta),\sin(\theta+d\theta))-(\cos\theta,\sin\theta)$
Well, it's tangent to the circle, and hence points in the perpendicular direction. And its length is $d\theta$. Hence it's $d\theta(-\sin\theta,\cos\theta)$.
[edit]
This turns out to be the same as Jyrki Lahtonen's proof B.
