[tex]\tan x=\dfrac13=\dfrac{\sin x}{\cos x}[/tex]
[tex]\tan x>0[/tex], but we're told that [tex]\cos x<0[/tex], which means [tex]\sin x<0[/tex]. So we should expect [tex]\csc x<0[/tex] as well.
Recall the Pythagorean identity:
[tex]\sin^2x+\cos^2x=1\implies\tan^2x+1=\sec^2x\implies\sec x=\pm\sqrt{\tan^2x+1}[/tex]
But we know [tex]\cos x<0[/tex], so [tex]\sec x<0[/tex], too.
[tex]\implies\sec x=-\sqrt{\tan^2x+1}=-\sqrt{\dfrac19+1}=-\sqrt{\dfrac{10}9}[/tex]
[tex]\implies\cos x=-\sqrt{\dfrac9{10}}[/tex]
[tex]\sin^2x+\cos^2x=1\implies\sin x=-\sqrt{1-\left(-\sqrt{\dfrac9{10}}\right)^2}=-\sqrt{\dfrac1{10}}[/tex]
[tex]\implies\csc x=-\sqrt{10}[/tex]
