well... simple enough, use the pythagorean identities
[tex]\bf \textit{Pythagorean Identities}
\\ \quad \\
sin^2(\theta)+cos^2(\theta)=1
\\ \quad \\
\boxed{1+cot^2(\theta)=csc^2(\theta)}
\\ \quad \\
1+tan^2(\theta)=sec^2(\theta)\\\\
-------------------------------\\\\
1+cot^2(\theta)=csc^2(\theta)\implies \pm\sqrt{1+cot^2(\theta )}=csc(\theta )[/tex]
so, which is it? the +/-? well, we know the angle is in the 4th quadrant
we also know cosecant is 1/sine.... so.. what's the sign of sine on the IV quadrant? well, is negative, if sine is negative, the cosecant is also negative, thus [tex]\bf -\sqrt{1+cot^2(\theta )}=csc(\theta )[/tex]
