University of South Carolina High School Math Contest/1993 Exam/Problem 18

Problem

The minimum value of the function

$\displaystyle f(x) = \frac{\sin (x)}{\sqrt{1 - \cos^2 (x)}} + \frac{\cos(x)}{\sqrt{1 - \sin^2 (x) }} + \frac{\tan(x)}{\sqrt{\sec^2 (x) - 1}} + \frac{\cot (x)}{\sqrt{\csc^2 (x) - 1}}$

as $x$ varies over all numbers in the largest possible domain of $f$ , is

$\mathrm{(A) \ }-4 \qquad \mathrm{(B) \ }-2 \qquad \mathrm{(C) \ }0 \qquad \mathrm{(D) \ }2 \qquad \mathrm{(E) \ }4$

Solution

Recall the Pythagorean Identities:

$\sin^2 x + \cos^2 x = 1$ $\tan^2 x + 1 = \sec^2 x$ $1 + \cot^2 x = \csc^2 x$

We can now simplify the function to

$f(x) = \frac{\sin(x)}{\pm \sin (x)}+\frac{\cos(x)}{\pm \cos(x)} + \frac{\tan(x)}{\pm \tan(x)} + \frac{\cot(x)}{\pm \cot(x)}$

which is just $\pm 1 \pm 1 \pm 1 \pm 1$ . The minimum value is thus -4.

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University of South Carolina High School Math Contest/1993 Exam/Problem 18

Problem

Solution