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Prove that : $\frac{\tan \theta}{1- \cot \theta} + \frac{\cot \theta}{1-\tan \theta} = 1 + \tan \theta + \cot \theta$
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$$\begin{aligned}& LHS = \frac{\tan \theta}{1 - \frac{1}{\tan \theta}} + \frac{\frac{1}{\tan \theta}}{1-\tan \theta} \\ & = \frac{\tan^2\theta}{\tan \theta - 1} - \frac{1}{\tan \theta(1 - \tan \theta)} \\ & = \frac{\tan^3\theta - 1}{\tan \theta(\tan \theta - 1)} \\ & = \frac{(\tan \theta - 1) (\tan^2\theta + \tan \theta+ 1)}{\tan \theta(\tan \theta - 1)} \\ & = \tan\theta + 1 + \cot\theta = RHS\end{aligned}$$