Answer
Please see below.
Work Step by Step
To find the slope of the tangent line at the point $(0,1)$, we should first find the slope of the secant line joining $(x, \cos x)$ and $(0,1)$ as follows.$$m=\frac{\Delta y}{\Delta x} \quad \Rightarrow \quad m=\frac{y_2-y_1}{x_2-x_1}=\frac{\cos x-1}{x-0}=\frac{\cos x -1}{x} .$$Now, to find the slope of the tangent line at the point $(0,1)$, we must find the limit of $m$ when $x$ approaches $0$; that is,$$\lim_{x \to 0} \frac{\cos x -1}{x}=\lim_{x \to 0}- \left (\frac{1- \cos x}{x} \right )=0$$(Please note that we have used Theorem 1.9 (2) in finding the limit).
