$\begin{align}  & \frac{1}{\tan \beta +\cot \beta }=\frac{1}{\frac{\sin \beta }{\cos \beta }+\frac{\cos \beta }{\sin \beta }} \\ & \frac{1}{\tan \beta +\cot \beta }=\frac{1}{\frac{{{\sin }^{2}}\beta +{{\cos }^{2}}\beta }{\sin \beta \cos \beta }} \\ & \frac{1}{\tan \beta +\cot \beta }=\frac{\sin \beta \cos \beta }{{{\sin }^{2}}\beta +{{\cos }^{2}}\beta }=\sin \beta \cos \beta  \\\end{align}$