1991 AHSME Problem 21

For all real numbers x except x=0 and x=1 the function f(x) is defined by f(x/(x-1))=1/x. Suppose 0\leq t\leq \pi/2. What is the value of f(\sec^2t)?

\textbf{(A) } \sin^2\theta\qquad \textbf{(B) } \cos^2\theta\qquad \textbf{(C) } \tan^2\theta\qquad \textbf{(D) } \cot^2\theta\qquad \textbf{(E) } \csc^2\theta

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.