If anyone could explain how the following is done, it would be greatly appreciated!

We have the following info about three functions A, B, and C for all values of T:

A(t))' = B(t)

(B(t))' = A(t)

(A(t))^2 - (B(t))^2 = 1

C(t) = B(t)/A(t)

From this, we can gather that the Maclaurin series for A(t) is 1 + t^2/2! + t^4/4! + t^6/6!... and the Maclaurin series for B(t) is t + t^3/3! + t^5/5! + t^7/7+...

a) What is the Maclaurin series for A(it)?

b) What is the Maclaurin series for B(it)?