Establishing Continuity of $\cos x$ based on Continuity of $\sin x$

Question

If we have established that $\lim_{x\to c} \sin x = \sin c$, is it enough to argue that $\cos x$ is just a translation of $\sin x$ in order to establish that $\lim_{x\to c} \cos x = \cos c$?

Answer 1

Yes since the composition of continuous function is continuous and we have

$$\cos x = \sin\left(\frac{\pi}2-x\right)$$

with $\frac{\pi}2-x$ continuous, we can conclude that $\cos x$ also is continuous.

Refer also to the related:

• composition of continuous functions
• Real Analysis: Continuity of a Composition Function