Here's an interactive diagram of how the six trig functions are defined. Click and drag to change the angle.
Now you can see where the names "tangent" and "secant" come from, as well as what the "co" prefix is doing in there. You can also see why everything except sine and cosine explode out to infinity under some conditions.
Compare the two views of tangent and cotangent; they're both straddling a pair of congruent triangles.
The alt view of cosecant is out of the way so it doesn't overlap secant, which is a shame, because it would neatly touch cotangent.
But then, maybe you can see how cotangent could be moved down with cosecant, too.
There are so many congruent and similar triangles in this diagram! See how many you can find, and what other patterns you notice.