2007/08 AHSMC Part 1 Problem 12

Problem 12 of 16

Let $\theta$ be an acute angle such that $\sec^2\theta + \tan^2\theta = 2$ . The value of $\csc^2\theta + \cot^2\theta$ is

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Solution

Let $s = \sin^2\theta$ and $c = \cos^2\theta$ . We have $\sec^2\theta + \tan^2\theta = \dfrac{1+s}{c} = 2$ . Since $s + c = 1$ , $2 - c = 2c$ so that $c = \dfrac{2}{3}$ . It follows that $s = \dfrac{1}{3}$ . Now $\csc^2\theta + \cot^2\theta = \dfrac{1+c}{s} = 5$ . The answer is (D).

University of Alberta

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