Chapter 2.5, Problem 45ES

Use Formula $\left(1\right)$ , the alternative form for the definition of derivative given in Formula $\left(13\right)$ of Section $2.2$ , that is,

$f^{\prime }\left(x\right)=\underset{w\to x}{\lim }\frac{f\left(w\right)-f\left(x\right)}{w-x}$

And the difference identity

$\sin \alpha -\sin \beta =2\sin \left(\frac{\alpha -\beta }{2}\right)\cos \left(\frac{\alpha +\beta }{2}\right)$

to show that $\frac{d}{dx}\left[\sin x\right]=\cos x$ .