Why is the secant function is an even function and the tangent and cosecant are odd functions?

I find it convenient to express other trigonometric functions in terms of sine and cosine - that tends to simplify things.

The secant function is even because it is the reciprocal of the cosine function, which is even.

The tangent function is the sine divided by the cosine - an odd function divided by an even function. Therefore it is odd. The cosecant is the reciprocal of an odd function, so it is naturally also an odd function.