How to find the minimum of $f(x)=(\sin(x)+\cos(x)+\tan(x)+\cot(x)+\sec(x)+\csc(x))^2$?
I need to find the minimum of $f(x)$ with $$f(x)=(\sin(x)+\cos(x)+\tan(x)+\cot(x)+\sec(x)+\csc(x))^2$$ Could you help me with some clues?
You can do the following:
Let $$\sin x + \cos x = y$$
Then we have that $$\tan x + \cot x = \frac{2}{y^2 -1}$$
and
$$\sec x + \csc x = \frac{2y}{y^2-1}$$
I believe $$\sin x + \cos x + \tan x + \cot x + \sec x + \csc x $$ simplifies to $$y + \frac{2}{y-1}$$ (but I tried doing it in my head, so might have made mistakes).
And you can use the standard calculus techniques, now. (but take care to eliminate the corner cases etc).