calculus - 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.

Answer

Consider the image from this "proof without words",

Asymptotically, the angle between the black radius and the red vertical line is complementary to both angles marked as $\theta$. Thus, asymptotically, those angles are equal, and the two red triangles are similar. Therefore, by similar triangles,

$$\frac{\mathrm{d}\sin(\theta)}{\mathrm{d}\theta}=\frac{\cos(\theta)}{1}$$
To get the derivative of $\cos(\theta)$, recall that $\cos(\theta)=\sin\left(\frac\pi2-\theta\right)$ and $\sin(\theta)=\cos\left(\frac\pi2-\theta\right)$. Then the Chain Rule says
$$\begin{align} \frac{\mathrm{d}}{\mathrm{d}\theta}\cos(\theta) &=\frac{\mathrm{d}}{\mathrm{d}\theta}\sin\left(\frac\pi2-\theta\right)\\ &=-\cos\left(\frac\pi2-\theta\right)\\[3pt] &=-\sin(\theta) \end{align}$$