$frac{sqrt{left(operatorname{cosec}^{2} A-1 ight)}}{operatorname{cosec} A}$ is equal to

$frac{sqrt{operatorname{cosec}^{2} A-1}}{operatorname{cosec} A}=frac{sqrt{cot ^{2} A}}{operatorname{cosec} A}=frac{cot A}{operatorname{cosec} A}$

$=frac{cos A}{sin A} imes sin A=cos A$

$[ herefore operatorname{cosec} A . sin A=1]$