Prove that: $\displaystyle (\operatorname{cosec} \theta-\sin \theta)(\sec \theta-\cos \theta)(\tan \theta+\cot \theta)=1$ This section consists of $\displaystyle 4$ questions of $\displaystyle 5$ marks each.

L.H.S. \(\displaystyle =\left(\frac{1}{\sin \theta}-\sin \theta\right)\left(\frac{1}{\cos \theta}-\cos \theta\right)\left(\frac{\sin \theta}{\cos \theta}+\frac{\cos \theta}{\sin \theta}\right)\) \[\begin{aligned} & =\left(\frac{1-\sin ^{2} \theta}{\sin \theta}\right)\left(\frac{1-\cos ^{2} \theta}{\cos \theta}\right)\left(\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\cos \theta \sin \theta}\right) \\ & =\frac{\cos ^{2} \theta}{\sin \theta} \times \frac{\sin ^{2} \theta}{\cos \theta} \times \frac{1}{\cos \theta \sin \theta} \\ & =1=\text { R.H.S } \end{aligned} \]