The most direct way to show this, in my opinion, is from the definition of cosine as adjacent over hypotenuse. Think of a circle of radius 1 centered at the origin. Draw a straight line (neither vertical nor horizontal) from the origin to a point on the circle in the first quadrant. Call its angle theta. From that intersection point, draw a vertical line down to the x axis. Now, the length of the x axis from the origin to the perpendicular you drew is the adjacent side, correct? It has to be labeled with a positive number, because x is positive in the first quadrant (so is y, but that's not relevant to the cosine function). The hypotenuse is always positive, because it's a length. Therefore, the cosine is positive. Finally, the angle theta is positive, because you're in the first quadrant.
Now flip your original straight line about the x axis. You now have a negative angle: -theta, which is in the fourth quadrant. The adjacent side is still positive, because x is always positive in the fourth quadrant, and the hypotenuse is always positive. Therefore, the cosine function will still be positive. In addition, its value will be the same as before.
You can repeat this argument in the second and third quadrants, where the cosine function is negative (since x is negative there), to convince yourself that, indeed, the cosine function is even.
A similar argument will show you, by the definition of the sine function, that it is odd.
Alternatively, you can show the result from the Taylor series expansions of the two functions. The cosine function has only even powers of x in its expansion, and the sine function has only odd powers of x in its expansion. The result follows immediately from that.
Does this make sense?