Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

$(\operatorname{cosec} \theta-\cot \theta)^{2}=\frac{1-\cos \theta}{1+\cos \theta}$

$(\operatorname{cosec} \theta-\cot \theta)^{2}=\frac{1-\cos \theta}{1+\cos \theta}$

$L.H.S.=(\operatorname{cosec} \theta-\cot \theta)^{2}$

$=\left(\frac{1}{\sin \theta}-\frac{\cos \theta}{\sin \theta}\right)^{2}$

$=\frac{(1-\cos \theta)^{2}}{(\sin \theta)^{2}}=\frac{(1-\cos \theta)^{2}}{\sin ^{2} \theta}$

$=\frac{(1-\cos \theta)^{2}}{1-\cos ^{2} \theta}=\frac{(1-\cos \theta)^{2}}{(1-\cos \theta)(1+\cos \theta)}=\frac{1-\cos \theta}{1+\cos \theta}$

$=$ $R.H.S.$