The tangent function

The domain of the tangent function is a little messy to describe, so let us look at the so-called Principal Tangent Function, ${\rm Tan}:(-\pi/2,\pi/2)\rightarrow(-\infty,\infty)$, with ${\rm Tan}(x) = \tan(x)$.The formula we derive here will hold for the general tangent function as well. We wish to compute

$${\rm Tan}'(a) \equiv \lim_{h\rightarrow 0}\frac{{\rm Tan}(a+h)-{\rm Tan}(a)}{h}$$

We employ a little algebra:

$$\begin{array} {rcl} {\displaystyle \frac{{\rm Tan}(a+h)-{\rm... ...aystyle \frac{\sin(h)}{h}\frac{1}{\cos(a+h)\cos(a)}}\end{array}$$

so that it is clear that

$$\lim_{h\rightarrow 0}\frac{{\rm Tan}(a+h)-{\rm Tan}(a)}{h} = 1\cdot\frac{1}{(\cos(a))^2} = \sec^2(a)$$

so ${\rm Tan}'(a) = \sec^2(a)$, and we show in the same fashion that $\tan'(a) = \sec^2(a)$. So, for example, the slope of the tangent line to $y=\tan(x)$ at (0,0) is $\sec^2(0) = 1$.