Question
Prove.
$(\operatorname{cosec} A-\sin A )(\sec A -\cos A )(\tan A +\cot A )= 1$
$(\operatorname{cosec} A-\sin A )(\sec A -\cos A )(\tan A +\cot A )= 1$
✓
Answer
$\text { LHS }=(\operatorname{cosec} A-\sin A )(\sec A -\cos A )(\tan A +\cot A ) $
$=\left(\frac{1}{\sin A}-\sin A\right)\left(\frac{1}{\cos A}-\cos A\right)\left(\frac{1}{\tan A}+\tan A\right) $
$ =\left(\frac{1-\sin ^2 A}{\sin A}\right)\left(\frac{1-\cos ^2 A}{\cos A}\right)\left(\frac{\sin A}{\cos A}+\frac{\cos A}{\sin A}\right)$
$ =\left(\frac{\cos ^2 A}{\sin A}\right)\left(\frac{\sin ^2 A}{\cos A}\right)\left(\frac{\sin ^2 A+\cos ^2 A}{\sin A \cdot \cos A}\right) $
$ =\left(\frac{\cos ^2 A}{\sin A}\right)\left(\frac{\sin ^2 A}{\cos A}\right)\left(\frac{1}{\sin A \cdot \cos A}\right) $
$=1=\text { RHS }$
$=\left(\frac{1}{\sin A}-\sin A\right)\left(\frac{1}{\cos A}-\cos A\right)\left(\frac{1}{\tan A}+\tan A\right) $
$ =\left(\frac{1-\sin ^2 A}{\sin A}\right)\left(\frac{1-\cos ^2 A}{\cos A}\right)\left(\frac{\sin A}{\cos A}+\frac{\cos A}{\sin A}\right)$
$ =\left(\frac{\cos ^2 A}{\sin A}\right)\left(\frac{\sin ^2 A}{\cos A}\right)\left(\frac{\sin ^2 A+\cos ^2 A}{\sin A \cdot \cos A}\right) $
$ =\left(\frac{\cos ^2 A}{\sin A}\right)\left(\frac{\sin ^2 A}{\cos A}\right)\left(\frac{1}{\sin A \cdot \cos A}\right) $
$=1=\text { RHS }$
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