i was wondering how you prove this equation

As usual with these types of problems your first step is to convert the tan, cot, sec, and csc functions into their sin and cos equivalents. Then it's a matter of simplifying and looking for opportunities to apply other well-known identities such as sin^2x + cos^2x = 1.

By the way, what you wrote is this:

tan x + \frac {cot x}{csc x} = sec x.

but I'm pretty sure what you actually meant is this:

\frac {tan x + cot x}{csc x} = sec x

Please use parentheses to avoid confusion.

Not a real equation but an identity.

Use your identities on the inside of the text cover and follow the other answer that was given.