## Calculus (3rd Edition)

$$G'(z)= -(\tan z -\cot z)^{-2}(\sec^2z+\csc^2 z).$$
Since $G(z)= \frac{1}{\tan z -\cot z}$, rewriting $G$ as follows $$G(z)= (\tan z -\cot z)^{-1}.$$ Now, the derivative is giving by $$G'(z)= -(\tan z -\cot z)^{-2}(\sec^2z+\csc^2 z).$$