Putnam inspired problem

Question

The following is a beautiful problem from Putnam 2003

minimize $|\sin x + \cos x + \tan x + \csc x + \sec x + \cot x|$

I was thinking about a small variation of the above problem

minimize $|\sin x + \cos x + \tan x - \csc x - \sec x - \cot x|$

Thanks.

Answer 1

Let $x=-\dfrac{\pi}{4}$. Then $\cos x = -\sin x, \sec x = -\csc x, \tan x = \cot x$ and the expression inside the absolute values is $0$.