$\begin{align}  & \frac{\sin t}{1-\cos t}=\frac{\sin t}{1-\cos t}\times \frac{1+\cos t}{1+\cos t} \\ & \frac{\sin t}{1-\cos t}=\frac{(\sin t+\sin t\cos t)}{1-{{\cos }^{2}}t} \\ & \frac{\sin t}{1-\cos t}=\frac{\sin t+\sin t\cos t}{{{\sin }^{2}}t} \\ & \frac{\sin t}{1-\cos t}=\frac{\sin t}{{{\sin }^{2}}t}+\frac{\sin t\cos t}{{{\sin }^{2}}t} \\ & \frac{\sin t}{1-\cos t}=\frac{1}{\sin t}+\frac{\cos t}{\sin t}=\csc t+\cot t \end{align}$