If f((x)/(x-1))=1x for all x{0,1}, and 0<θ<(π)/(2), then f(sec^2θ)=

1991 AMC 12 Problem 21

Attempt Problem 21 of the 1991 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1991 AMC 12 solutions, or check the answer key.

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21.

f\left(\frac{x}{x-1}\right)=\frac1x

for all

x\notin\{0,1\},

and

0\lt\theta\lt\frac{\pi}{2},

then

f(\sec^2\theta)=

\sin^2\theta

\cos^2\theta

\tan^2\theta

\cot^2\theta

\csc^2\theta

Choose

x

so that

\frac{1}{x}=\sec^2\theta

Use

1-\cos^2\theta=\sin^2\theta

Since

\sec^2\theta=\frac1{\cos^2\theta} =\frac{\csc^2\theta}{\csc^2\theta-1},

choose

x=\csc^2\theta

in the definition. Then

f(\sec^2\theta)=\frac1{\csc^2\theta}=\sin^2\theta.

Thus the correct answer is A.