This is from the Putnam exam, 2003.

"Find the minimum value of $\displaystyle |\sin(x)+\cos(x)+\tan(x)+\cos(x)+\sec(x)+\csc(x)|$ for all real x".

My thoughts:

First I rewrote everything in terms of sine and cosine. After simplifying I get:

$\displaystyle \sin^2(x)\cos(x)+\sin(x)\cos^2(x)+\sin(x)\cos(x)+1$

Now here's my thinking... if I let a=sin(x) and b=cos(x) then I can use Lagrange Multipliers to find the minimum. What's the constraint though?