7.16 If, for all real numbers $x \neq 0,1$,
$$
f\left(\frac{x}{x-1}\right)=\frac{1}{x} \text {, }
$$

and $0<\theta<\frac{\pi}{2}$, then $f\left(\sec ^{2} \theta\right)$ equals
(A) $\sin ^{2} \theta$.
(B) $\cos ^{2} \theta$.
(C) $\operatorname{tg}^{2} \theta$.
(D) $\operatorname{ctg}^{2} \theta$.
(E) $\csc ^{2} \theta$.
(42nd American High School Mathematics Examination, 1991)
[Solution] Let $t=\frac{x}{x-1}$, then $x=\frac{t}{t-1}$. According to the problem, we have
$$
f(t)=\frac{t-1}{t}=1-\frac{1}{t}.
$$

Thus, $f\left(\sec ^{2} \theta\right)=1-\frac{1}{\sec ^{2} \theta}=1-\cos ^{2} \theta=\sin ^{2} \theta$. Therefore, the answer is $(A)$.