Answer
$$\frac{dy}{dt}=\csc^2(t/2)\Big(1+\cot(t/2)\Big)^{-3}$$
Work Step by Step
$$\frac{dy}{dt}=\frac{d}{dt}\Big(1+\cot(t/2)\Big)^{-2}$$
$$\frac{dy}{dt}=-2\Big(1+\cot(t/2)\Big)^{-3}\frac{d}{dt}(1+\cot(t/2))$$
$$\frac{dy}{dt}=-2\Big(1+\cot(t/2)\Big)^{-3}\Big(0-\csc^2(t/2)\frac{d}{dt}(t/2)\Big)$$
$$\frac{dy}{dt}=-2\Big(1+\cot(t/2)\Big)^{-3}\Big(-\frac{1}{2}\csc^2(t/2)\Big)$$
$$\frac{dy}{dt}=\csc^2(t/2)\Big(1+\cot(t/2)\Big)^{-3}$$