^2 means to (second power)
I would really love to know exactly how to do it I'm completely confused

The directions say verify each identity

Remember that tan(x) = sin(x)/cos(x), so if you make that substitution you get:


\cos^2x + ( \frac {\sin^2x}{\cos^2x} ) \cos^2x = \cos^2x + \sin^2x


Now you apply the basic identity


\cos^2x + \sin^2x = 1


In general for proving trig identities like this you should always replace any tangent, cortangent, secant and cosecant functions with their sine and cosine equivalents, and if you remember a few key identites it is usually pretty straight forward to work out.

ebaines is exceptional at math among other things, will help greatly.
I googled your formula and found this, seems helpful:
Trigonometric Identities (http://www.math.com/tables/trig/identities.htm)