Prove the following identities (1-16)

Prove the following identities (1-16)

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


$\mathrm{LHS}=\operatorname{cosec} x(\sec x-1)-\cot x(1-\cos x)$

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

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

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

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

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

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

$=\frac{\sin x}{\cos x}-\sin x$

$=\tan x-\sin x$


Hence proved.


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