Think of poet William Blake.
Then, think of that Blake poem The Tyger in all its stylistic, symmetrical, metered wonderfulness. After this dose of poetic license, you'll definitely be in the mood for trig functions and their fearful symmetry.
We can figure out the symmetry of the trig functions by comparing their values in Quadrant I and Quadrant IV.
Start with a representative triangle in Quadrant I.
In Quadrant I, 
That's just a rehashing of our basic trig ratios.
Now, let's look at the same triangle flipped into Quadrant IV.
In Quadrant IV, 
They're the same ratios as before, except we've got a negative y-value.
Therefore sin(-ɵ) = -sin ɵ, making sine an odd function.
And cos(-ɵ) = cos ɵ, making cosine an even function.
Finally, tan(-ɵ) = -tan ɵ, making tangent an odd function, too.
How 'bout the reciprocal functions? Since cosecant (csc) is the reciprocal of sine, cosecant is also an odd function.
Since secant (sec) is the reciprocal of cosine, secant is also an even function.
Last up, since cotangent (cot) is the reciprocal of tangent, cotangent is also an odd function.
That's all she wrote.