Since you posted in the Pre-Calculus forum, I suppose you are to use the definition:
$$f'(x)\equiv\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$
So, for $f(x)=\sin(x)$, we have:
$$f'(x)\equiv\lim_{h\to0}\frac{\sin(x+h)-\sin(x)}{h}$$
Using the angle-sum identity for the sine function, we may write:
$$f'(x)=\lim_{h\to0}\frac{\sin(x)(\cos(h)-1)+\cos(x)\sin(h)}{h}$$
$$f'(x)=\sin(x)\lim_{h\to0}\frac{\cos(h)-1}{h}+\cos(x)\lim_{h\to0}\frac{\sin(h)}{h}$$
$$f'(x)=\sin(x)\lim_{h\to0}\frac{\cos^2(h)-1}{h(\cos(h)+1)}+\cos(x)\lim_{h\to0}\frac{\sin(h)}{h}$$
Using a Pythagorean identity on the first limit on the RHS, we obtain:
$$f'(x)=-\sin(x)\lim_{h\to0}\frac{\sin^2(h)}{h(\cos(h)+1)}+\cos(x)\lim_{h\to0}\frac{\sin(h)}{h}$$
$$f'(x)=-\sin(x)\lim_{h\to0}\frac{\sin(h)}{h}\cdot\lim_{h\to0}\frac{\sin(h)}{(\cos(h)+1)}+\cos(x)\lim_{h\to0}\frac{\sin(h)}{h}$$
Using the fact that $$\lim_{u\to0}\frac{\sin(u)}{u}=1$$, we have:
$$f'(x)=-\sin(x)(1)(0)+\cos(x)(1)=\cos(x)$$
Can you now obtain a result for $f(x)=\cos(x)$?