Question

Perform the subtraction and use the fundamental identities to simplify. cotx-(csc^2x)/(cotx)

Answer 1

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

$\bf cot(x)-\left[ \cfrac{1}{cot(x)}+\cfrac{cot^2(x)}{cot(x)} \right]\implies cot(x)-\left[ tan(x)+cot(x) \right] \\\\\\ cot(x)-tan(x)-cot(x)\implies -tan(x)$