Prove that (tanA + cotA) (cosecA - sinA) (secA - cosA) = 1 - Vidyadip

Question

Prove that
(tanA + cotA) (cosecA - sinA) (secA - cosA) = 1

✓

Answer

$(\tan A+\cot A)(\operatorname{cosec} A-\sin A)(\sec A-\cos A)$
$=\left(\frac{\sin A}{\cos A}+\frac{\cos A}{\sin A}\right)\left(\frac{1}{\sin A}-\sin A\right)\left(\frac{1}{\cos A}-\cos A\right)$
$=\left(\frac{\sin ^2 A+\cos ^2 A}{\sin A \cos A}\right)\left(\frac{1-\sin ^2 A}{\sin A}\right)\left(\frac{1-\cos ^2 A}{\cos A}\right)$
$=\left(\frac{1}{\sin A \cos A}\right)\left(\frac{\cos ^2 A}{\sin A}\right)\left(\frac{\sin ^2 A}{\cos A}\right)$
$=1$

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