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Prove that :
$(\text{cosec } \theta - \sin \theta) (\sec \theta - \cos \theta) (\tan \theta + \cot \theta) = 1$
$(\text{cosec } \theta - \sin \theta) (\sec \theta - \cos \theta) (\tan \theta + \cot \theta) = 1$
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Sol.
L.H.S.=$(\frac{1}{\sin \theta} - \sin \theta) (\frac{1}{\cos \theta} - \cos \theta) (\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta})$
$= (\frac{1-\sin^2 \theta}{\sin \theta}) (\frac{1-\cos^2 \theta}{\cos \theta}) (\frac{\sin^2 \theta+\cos^2 \theta}{\cos \theta \sin \theta})$
$= (\frac{\cos^2 \theta}{\sin \theta}) \times (\frac{\sin^2 \theta}{\cos \theta}) \times (\frac{1}{\cos \theta \sin \theta})$
$=1 = \text{R.H.S}$
L.H.S.=$(\frac{1}{\sin \theta} - \sin \theta) (\frac{1}{\cos \theta} - \cos \theta) (\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta})$
$= (\frac{1-\sin^2 \theta}{\sin \theta}) (\frac{1-\cos^2 \theta}{\cos \theta}) (\frac{\sin^2 \theta+\cos^2 \theta}{\cos \theta \sin \theta})$
$= (\frac{\cos^2 \theta}{\sin \theta}) \times (\frac{\sin^2 \theta}{\cos \theta}) \times (\frac{1}{\cos \theta \sin \theta})$
$=1 = \text{R.H.S}$