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What is the value of $(\sec \theta-\cos \theta)(\operatorname{cosec} \theta-\sin \theta)(\cot \theta+\tan \theta) ?$
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The given expression is:
$(\sec \theta-\cos \theta)(\operatorname{cosec} \theta-\sin \theta)(\cot \theta+\tan \theta)$
$=\left(\frac{1}{\cos \theta}-\cos \theta\right)\left(\frac{1}{\sin \theta}-\sin \theta\right)\left(\frac{\cos \theta}{\sin \theta}+\frac{\sin \theta}{\cos \theta}\right)$
$=\left(\frac{1-\cos ^{2} \theta}{\cos \theta}\right)\left(\frac{1-\sin ^{2} \theta}{\sin \theta}\right)\left(\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\sin \theta \cos \theta}\right)$
$=\frac{\sin ^{2} \theta}{\cos \theta} \cdot \frac{\cos ^{2} \theta}{\sin \theta} \times \frac{1}{\sin \theta \cos \theta}=\frac{\sin ^{2} \theta \cdot \cos ^{2} \theta}{\cos ^{2} \theta \cdot \sin ^{2} \theta}=1$
$(\sec \theta-\cos \theta)(\operatorname{cosec} \theta-\sin \theta)(\cot \theta+\tan \theta)$
$=\left(\frac{1}{\cos \theta}-\cos \theta\right)\left(\frac{1}{\sin \theta}-\sin \theta\right)\left(\frac{\cos \theta}{\sin \theta}+\frac{\sin \theta}{\cos \theta}\right)$
$=\left(\frac{1-\cos ^{2} \theta}{\cos \theta}\right)\left(\frac{1-\sin ^{2} \theta}{\sin \theta}\right)\left(\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\sin \theta \cos \theta}\right)$
$=\frac{\sin ^{2} \theta}{\cos \theta} \cdot \frac{\cos ^{2} \theta}{\sin \theta} \times \frac{1}{\sin \theta \cos \theta}=\frac{\sin ^{2} \theta \cdot \cos ^{2} \theta}{\cos ^{2} \theta \cdot \sin ^{2} \theta}=1$
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