Question about derivative of $\cos(x)$.

\begin{align} & \lim_{h\to0}\frac{(\cos x)(\cos h)-(\sin x)(\sin h) -\cos x}h \\[12pt] = {} & (\cos x)\,\, \underbrace{\lim_{h\to0} \left( \frac{(\cos h) -1} h \right)}_\text{first limit} - (\sin x)\,\, \underbrace{\lim_{h\to0} \left( \frac{\sin h} h \right)}_\text{second limit} \end{align} The first limit is $0$ and the second is $1$.

That's the way to do this. He shouldn't be letting $h=0$ in the numerator while leaving $h$ in the denominator. The fact that the first limit is $0$ cannot be shown merely by showing that the limit of the numerator is $0.$ If that were valid, the the same argument argument would show that the second limit is $0,$ and that is wrong.