TITLE: Solve $\sin x+\cos x+\tan x+\csc x+\sec x+\cot x=-2$ in the interval $0<x<2\pi$
QUESTION [5 upvotes]: Solve $$\sin x+\cos x+\tan x+\csc x+\sec x+\cot x=-2$$ from $0<x<2\pi$.

Could you find the most elegant solution? Does it factorize?

REPLY [4 votes]: Substitute $\sin x + \cos x=t$
Also, $t^2=1+\sin 2x$
Given reduces to : 
$$\sin x+\cos x + \frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}+\frac{1}{\sin x}+\frac{1}{\cos x}$$
$$t+\frac{2}{t^2-1}+\frac{2t}{t^2-1}$$
$$t^3-t+2+2t=-2t^2+2$$
$$t^3+2t^2+t=0$$
$$t(t+1)^2=0$$
This is easy peasy.
Now you can easily solve the equation.
Then it's all about solving a nice trigonometry equation. You can use:$\frac{t}{\sqrt 2}=\sin(\frac{\pi}{4}+x)$