Question

What is $\cot \left(\frac{\theta }{2}\right)$ in terms of trigonometric functions?

Answer 1

$\cot \left(\frac{\theta }{2}\right)$ in terms of trigonometric functions:

Multiple Formulas for $\cot \left(\frac{\theta }{2}\right)$ are:

$$\begin{gathered} \cot \left(\frac{\theta }{2}\right)=\mathrm{cosec}\theta +\cot \theta \\ \cot \left(\frac{\theta }{2}\right)=\frac{1+\cos \theta }{\sin \theta } \\ \cot \left(\frac{\theta }{2}\right)=\pm \sqrt{\frac{1+\cos \theta }{1-\cos \theta }} \\ \cot \left(\frac{\theta }{2}\right)=\frac{\sin \theta }{1-\cos \theta } \end{gathered}$$

Therefore, $\cot \left(\frac{\theta }{2}\right)$ in terms of trigonometric functions is:

$$\begin{gathered} \mathit{c}\mathit{o}\mathit{t}\left(\frac{\theta }{2}\right)\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{e}\mathit{c}\mathbf{}\mathit{\theta }\mathbf{+}\mathit{c}\mathit{o}\mathit{t}\mathbf{}\mathit{\theta } \\ \mathit{c}\mathit{o}\mathit{t}\left(\frac{\theta }{2}\right)\mathbf{=}\frac{\mathbf{1}\mathbf{+}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{\theta }}{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{\theta }} \\ \mathit{c}\mathit{o}\mathit{t}\left(\frac{\theta }{2}\right)\mathbf{=}\mathbf{\pm }\sqrt{\frac{\mathbf{1}\mathbf{+}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{\theta }}{\mathbf{1}\mathbf{-}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{\theta }}} \\ \mathit{c}\mathit{o}\mathit{t}\left(\frac{\theta }{2}\right)\mathbf{=}\frac{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{\theta }}{\mathbf{1}\mathbf{-}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{\theta }} \end{gathered}$$