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