For eliminating the parameter:
Get in terms of . Then substitute that result into the second equation.
Alternately, get in terms of . Then substitute that result into the first equation.
You can do it this way: imagine a right triangle with "opposite side" of length x and "near side" of length 1. Then or . and secant is defined as "hypotenuse over near side". You can get the hypotenuse from the Pythagorean theorem.
That's really a "visual" way of getting the following identity:
You know that . Divide both sides by to get which is the same as .
Taking in that gives gives . Therefore, .