Douglas Wilhelm Harder, M.Math., LEL

If you've ever wondered why, $\sec(\theta) = \frac{1}{\cos(\theta)}$ or why $1 + \tan^2(\theta) = \sec^2(\theta)$, please read this document:

Trigonometry made somewhat easier.

It only requires you to understand the basic properties of similar right triangles: suppose we have two right-angled triangles $abc$ and $def$ (that is, triangles with sides of length $a$, $b$ and $c$; and $d$, $e$ and $f$; respectively), as shown in Figure 1.

Figure 1. Two similar triangles $abc$ and $def$.

First, because they are both right-angled triangles, we have that the Pythagorean theorem holds for each:

$a^2 + b^2 = c^2$ and $d^2 + e^2 = f^2$.

However, because these triangles are similar (all three angles are equal), we also have that the three equations

$\frac{a}{b} = \frac{d}{e}$, $\frac{a}{c} = \frac{d}{f}$ and $\frac{b}{c} = \frac{e}{f}$,

as well as their reciprocals also hold. Also,

$\frac{a}{d} = \frac{b}{e} = \frac{c}{f}$.

It also helps if you understand that the word tangent is derived from the Latin verb tangere meaning "to touch" and secant is derived from the Latin verb secare meaning "to cut".