Strangely worded, but I think they want you to express y as a function of x. For example in the first one if you use the identity you can get an expression for y(x) which has no term in it.
Trigonometric identities needed for these 2 questions:
(a) cos^2(theta) + sin^2(theta) = 1. With that in mind, cos(theta) = x/a and sin(theta) = y/b. That's how you have (x/a)^2+(y/b)^2 = 1
(a) tan^2(theta) + 1 = sec^2(theta). With that in mind, tan(theta) = x/a and sec(theta) = y/b. Then you have (x/a)^2 + 1 = (y/b)^2.