#### Answer

$\frac{csc^2(\frac{t}{2})}{(1+cot(\frac{t}{2}))^3}$

#### Work Step by Step

y=$(1+cot(\frac{t}{2}))^{-2}$
Apply the chain rule:
$\frac{dy}{dt}=-2(1+cot(\frac{t}{2}))^3.\frac{d}{dt}(1+cot(\frac{t}{2}))$
=$-2(1+cot(\frac{t}{2}))^{-3}.(-csc^2(\frac{t}{2})).\frac{d}{dt}(\frac{t}{2})$
=$\frac{csc^2(\frac{t}{2})}{(1+cot(\frac{t}{2}))^3}$