Prove the following identities (1-16)


Prove the following identities (1-16)

$(\operatorname{cosec} x-\sin x)(\sec x-\cos x)(\tan x+\cot x)=1$


$\mathrm{LHS}=(\operatorname{cosec} x-\sin x)(\sec x-\cos x)(\tan x+\cot x)$

$=\left(\frac{1}{\sin x}-\sin x\right)\left(\frac{1}{\cos x}-\cos x\right)\left(\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}\right)$

$=\left(\frac{1-\sin ^{2} x}{\sin x}\right)\left(\frac{1-\cos ^{2} x}{\cos x}\right)\left(\frac{\sin ^{2} x+\cos ^{2} x}{\cos x \sin x}\right)$

$=\left(\frac{\cos ^{2} x}{\sin x}\right)\left(\frac{\sin ^{2} x}{\cos x}\right)\left(\frac{1}{\cos x \sin x}\right)$



Hence proved.

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