Question

Find the minimum value of $\frac{\tan \theta -\cot \theta +1}{\tan \theta +\cot \theta -1}$

Answer 1

$\begin{array}{c}\frac{\tan \theta -\cot \theta +1}{\tan \theta +\cot \theta -1}\\ \Rightarrow \frac{\left({\sin }^2\theta -{\cos }^2\theta +\sin \theta .\cos \theta \right)}{\left({\sin }^2\theta +{\cos }^2\theta -\sin \theta .\cos \theta \right)}\\ \Rightarrow Rangeof\sin \theta =\left[-1,1\right]\\ \Rightarrow Rangeof\cos \theta =\left[-1,1\right]\\ MinimumValueis=\frac{0-0+1}{0+0-1}=\left(-1\right)\end{array}$