Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
$(\operatorname{cosec} A-\sin A)(\sec A-\cos A)=\frac{1}{\tan A+\cot A}$
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
$(\operatorname{cosec} A-\sin A)(\sec A-\cos A)=\frac{1}{\tan A+\cot A}$
$(\operatorname{cosec} A-\sin A)(\sec A-\cos A)=\frac{1}{\tan A+\cot A}$
$L.H.S.=(\operatorname{cosec} A-\sin A)(\sec A-\cos A)$
$=\left(\frac{1}{\sin A}-\sin A\right)\left(\frac{1}{\cos A}-\cos A\right)$
$=\left(\frac{1-\sin ^{2} A}{\sin A}\right)\left(\frac{1-\cos ^{2} A}{\cos A}\right)$
$=\frac{\left(\cos ^{2} A\right)\left(\sin ^{2} A\right)}{\sin A \cos A}$
$=\sin A \cos A$
$R.H.S=\frac{1}{\tan A+\cot A}$
$=\frac{1}{\frac{\sin A}{\cos A}+\frac{\cos A}{\sin A}}=\frac{1}{\sin ^{2} A+\cos ^{2} A}{\sin A \cos A}$
$=\frac{\sin A \cos A}{\sin ^{2} A+\cos ^{2} A}=\sin A \cos A$
Hence,$L . H . S=R . H . S$
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