Answer
See explanations.
Work Step by Step
a. For $f(x)=sin(x)$
Step 1. To prove that $f(x)=sin(x)$ is continuous at every point $x=c$, we need to show that $\lim_{h\to0}f(h+c)=f(c)$ (see Exercise 69)
Step 2. Use the Identity $sin(h+c)=sin(h)cos(c)+cos(h)sin(c)$; we have $\lim_{h\to0}f(h+c)=\lim_{h\to0}(sin(h)cos(c)+cos(h)sin(c))=sin(0)cos(c)+cos(0)sin(c)=sin(c)=f(c)$
Thus, we end our proof.
b. Similarly, for $g(x)=cos(x)$
Step 1. To prove that $g(x)=cos(x)$ is continuous at every point $x=c$, we need to show that $\lim_{h\to0}g(h+c)=g(c)$ (see Exercise 69)
Step 2. Use the Identity $cos(h+c)=cos(h)cos(c)-sin(h)sin(c)$, we have $\lim_{h\to0}g(h+c)=\lim_{h\to0}(cos(h)cos(c)-sin(h)sin(c))=cos(0)cos(c)-sin(0)sin(c)=cos(c)=g(c)$
Thus, we end our proof.
