Why are $\sin,\cos,\tan$ continuous

Question

I'm done with two courses in Analysis, but just can't seem to work out how I'll show the base trigonometric functions to be continuous.

Any references or indications for a simple, preferably elementary proof ?

Is it possible to do it relying only on $\epsilon$-$\delta$ arguments?

Answer 1

For example, using the fact that $$\cos(p)-\cos(q)=-2\sin\left(\frac{a+b}{2}\right)\sin\left(\frac{a-b}{2}\right),$$ yields $$|\cos(x+h)-\cos(x)|=2\left|\sin\left(\frac{2x+h}{2}\right)\sin\left(\frac{h}{2}\right)\right|\leq |h|.$$

If $\varepsilon >0$, take $\delta =\varepsilon $ and thus $$|h|\leq \delta \implies |\cos(x+h)-\cos(x)|\leq \varepsilon .$$

This prove the continuity of the cosine function. Do the same with the sine function gives the wished result.