Show that $\left(\text{cosec}\theta - \sin \theta\right) \left(\sec\theta - \cos\theta\right) \left(\tan\theta + \cot\theta\right) = 1$
$ \begin{aligned} \text{LHS} & = \left(\text{cosec}\theta - \sin \theta\right) \left(\sec\theta - \cos\theta\right) \left(\tan\theta + \cot\theta\right) \\ & = \left(\dfrac{1}{\sin\theta} - \sin\theta\right) \left(\dfrac{1}{\cos\theta} - \cos\theta\right) \left(\dfrac{\sin\theta}{\cos\theta} + \dfrac{\cos\theta}{\sin\theta}\right) \\ & = \left(\dfrac{1 - \sin^2 \theta}{\sin\theta}\right) \left(\dfrac{1 - \cos^2 \theta}{\cos\theta}\right) \left(\dfrac{\sin^2 \theta + \cos^2 \theta}{\sin\theta \cos\theta}\right) \\ & = \dfrac{\cos^2 \theta}{\sin\theta} \times \dfrac{\sin^2 \theta}{\cos\theta} \times \dfrac{1}{\sin\theta \cos\theta} \\ & = 1 \\ & = \text{RHS} \end{aligned} $