\(\bf cot(x)-\cfrac{csc^2(x)}{cot(x)}\\\\
-----------------------------\\\\
recall\qquad
\begin{array}{llll}
\textit{Pythagorean Identities}
\\ \quad \\
sin^2(\theta)+cos^2(\theta)=1
\\ \quad \\
\boxed{1+cot^2(\theta)=csc^2(\theta)}
\\ \quad \\
1+tan^2(\theta)=sec^2(\theta)
\end{array}\\\\
-----------------------------\\\\
cot(x)-\left[ \cfrac{1+cot^2(x)}{cot(x)} \right]\impliedby \textit{distributing the denominator}
\\\\\\\)
\(\bf cot(x)-\left[ \cfrac{1}{cot(x)}+\cfrac{cot^2(x)}{cot(x)} \right]\implies cot(x)-\left[ tan(x)+cot(x) \right]
\\\\\\
cot(x)-tan(x)-cot(x)\implies -tan(x)\)
