To see where this comes from, lets look at the graphs of sine, cosine and tangnt all plotted on the same set of axes. Sine is red, cosine is green, and tangent is gold.
Zoom in near zero. 
, so 
near zero looks just like 
. Notice how the red and gold curves come together near 0.
Similarly we could zoom in near 
. 
, so tangent looks like 
near 
.
When cosine is zero tangent blows up to infinity. Zoom in near 
. To the left of 
sine and cosine are both positive, so tangent is also positive and goes off to 
as we approach 
.
Immediately to the right of 
sine is still positive but cosine is negative, so tangent is also negative. As we approach 
from the right tangent goes to 
.
. Let's look at the graph of secant compared to the graph of cosine.
When cosine is 1 or -1 secant=cosine, so we see the two graphs touching exactly at those points.
Also notice that when cosine is positive, secant is positive, and when cosine is negative, secant is negative.
