$\begin{align}  & \int_{\tfrac{\pi }{3}}^{\tfrac{\pi }{2}}{\frac{\cos xdx}{1-\cos x}} \\ & \text{Using long division} \\ & 1-\cos x\overset{-1}{\overline{\left){\begin{align}  & \cos x \\ & \underline{\cos x-1} \\\end{align}}\right.}} \\ & \text{                          }1 \\ & \frac{\cos x}{1-\cos x}=-1+\frac{1}{1-\cos x} \\ & \int_{\tfrac{\pi }{3}}^{\tfrac{\pi }{2}}{\frac{\cos xdx}{1-\cos x}}=\int_{\tfrac{\pi }{3}}^{\tfrac{\pi }{2}}{\left( -1+\frac{1}{1-\cos x} \right)dx} \\ & \int_{\tfrac{\pi }{3}}^{\tfrac{\pi }{2}}{\frac{\cos xdx}{1-\cos x}}=\int_{\tfrac{\pi }{3}}^{\tfrac{\pi }{2}}{\left( -1+\frac{1}{1-\left( 1-2{{\sin }^{2}}\tfrac{x}{2} \right)} \right)}dx \\ & \int_{\tfrac{\pi }{3}}^{\tfrac{\pi }{2}}{\frac{\cos xdx}{1-\cos x}}=\int_{\tfrac{\pi }{3}}^{\tfrac{\pi }{2}}{\left( -1+\frac{1}{2{{\sin }^{2}}\tfrac{x}{2}} \right)dx} \\ & \int_{\tfrac{\pi }{3}}^{\tfrac{\pi }{2}}{\frac{\cos xdx}{1-\cos x}}=\int_{\tfrac{\pi }{3}}^{\tfrac{\pi }{2}}{\left( -1+\tfrac{1}{2}{{\operatorname{cosec}}^{2}}\tfrac{x}{2} \right)dx} \\ & \int_{\tfrac{\pi }{3}}^{\tfrac{\pi }{2}}{\frac{\cos xdx}{1-\cos x}}=\left[ -x-\frac{1}{2}\cot \tfrac{x}{2} \right]_{\tfrac{\pi }{3}}^{\tfrac{\pi }{2}} \\ & \int_{\tfrac{\pi }{3}}^{\tfrac{\pi }{2}}{\frac{\cos xdx}{1-\cos x}}=\left( -\tfrac{\pi }{2}+\tfrac{1}{2}\cot \tfrac{\pi }{4} \right)-\left( -\tfrac{\pi }{3}-\tfrac{1}{2}\cot \tfrac{\pi }{6} \right) \\ & \int_{\tfrac{\pi }{3}}^{\tfrac{\pi }{2}}{\frac{\cos xdx}{1-\cos x}}=-\frac{\pi }{2}+\frac{\pi }{3}-\frac{1}{2}+\frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{2}-\frac{1}{2}-\frac{\pi }{6} \\\end{align}$