University of South Carolina High School Math Contest/1993 Exam/Problem 18
Problem
The minimum value of the function
![$\displaystyle f(x) = \frac{\sin (x)}{\sqrt{1 - \cos^2 (x)}} + \frac{\cos(x)}{\sqrt{1 - \sin^2 (x) }} + \frac{\tan(x)}{\sqrt{\sec^2 (x) - 1}} + \frac{\cot (x)}{\sqrt{\csc^2 (x) - 1}}$](http://latex.artofproblemsolving.com/c/6/a/c6a4e021d142f4bad5a913725ac6e8f1316b9066.png)
as varies over all numbers in the largest possible domain of
, is
![$\mathrm{(A) \ }-4 \qquad \mathrm{(B) \ }-2 \qquad \mathrm{(C) \ }0 \qquad \mathrm{(D) \ }2 \qquad \mathrm{(E) \ }4$](http://latex.artofproblemsolving.com/2/f/7/2f7612b8e989eded4129ff00a31811dee6a6ef5e.png)
Solution
Recall the Pythagorean Identities:
![$\sin^2 x + \cos^2 x = 1$](http://latex.artofproblemsolving.com/a/9/5/a958fb1f7b5b87ef62ed4a2205643ef36cb3df4d.png)
![$\tan^2 x + 1 = \sec^2 x$](http://latex.artofproblemsolving.com/5/3/f/53fffbe6880b6565c56fbfa616bcbf130036e766.png)
![$1 + \cot^2 x = \csc^2 x$](http://latex.artofproblemsolving.com/2/2/b/22b3b506380cec4206c8a8c63c9397a1295d990d.png)
We can now simplify the function to
![$f(x) = \frac{\sin(x)}{\pm \sin (x)}+\frac{\cos(x)}{\pm \cos(x)} + \frac{\tan(x)}{\pm \tan(x)} + \frac{\cot(x)}{\pm \cot(x)}$](http://latex.artofproblemsolving.com/d/a/2/da2c2d95c37a5877c4614511357ef86499565d20.png)
which is just . The minimum value is thus -4.