Answer
$$ G'(z)= -(\tan z -\cot z)^{-2}(\sec^2z+\csc^2 z).$$
Work Step by Step
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
$$ G'(z)= -(\tan z -\cot z)^{-2}(\sec^2z+\csc^2 z).$$
