Prove the following identity,where the angles | vedclass

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(A) $L.H.S. = \frac{	an 	heta}{1-\cot 	heta} + \frac{\cot 	heta}{1-	an 	heta}$$= \frac{\frac{\sin 	heta}{\cos 	heta}}{1-\frac{\cos 	heta}{\si

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(A) $L.H.S. = \frac{	an 	heta}{1-\cot 	heta} + \frac{\cot 	heta}{1-	an 	heta}$$= \frac{\frac{\sin 	heta}{\cos 	heta}}{1-\frac{\cos 	heta}{\sin 	heta}} + \frac{\frac{\cos 	heta}{\sin 	heta}}{1-\frac{\sin 	heta}{\cos 	heta}}$$= \frac{\frac{\sin 	heta}{\cos 	heta}}{\frac{\sin 	heta-\cos 	heta}{\sin 	heta}} + \frac{\frac{\cos 	heta}{\sin 	heta}}{\frac{\cos 	heta-\sin 	heta}{\cos 	heta}}$$= \frac{\sin^2 	heta}{\cos 	heta(\sin 	heta-\cos 	heta)} - \frac{\cos^2 	heta}{\sin 	heta(\sin 	heta-\cos 	heta)}$$= \frac{1}{(\sin 	heta-\cos 	heta)} \left[ \frac{\sin^2 	heta}{\cos 	heta} - \frac{\cos^2 	heta}{\sin 	heta} ight]$$= \left( \frac{1}{\sin 	heta-\cos 	heta} ight) \left[ \frac{\sin^3 	heta-\cos^3 	heta}{\sin 	heta \cos 	heta} ight]$$= \left( \frac{1}{\sin 	heta-\cos 	heta} ight) \left[ \frac{(\sin 	heta-\cos 	heta)(\sin^2 	heta+\cos^2 	heta+\sin 	heta \cos 	heta)}{\sin 	heta \cos 	heta} ight]$$= \frac{1+\sin 	heta \cos 	heta}{\sin 	heta \cos 	heta}$$= \frac{1}{\sin 	heta \cos 	heta} + \frac{\sin 	heta \cos 	heta}{\sin 	heta \cos 	heta}$$= \operatorname{cosec} 	heta \sec 	heta + 1 = R.H.S.$

Prove the following identity,where the angles involved are acute angles for which the expressions are defined:
$\frac{\tan \theta}{1-\cot \theta}+\frac{\cot \theta}{1-\tan \theta}=1+\sec \theta \operatorname{cosec} \theta$

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Prove the following identity,where the angles involved are acute angles for which the expressions are defined:$\frac{\tan \theta}{1-\cot \theta}+\frac{\cot \theta}{1-\tan \theta}=1+\sec \theta \operatorname{cosec} \theta$