Prove the Identity : \( \cosec x(\sec x-1)-\cot x(1-\cos x)=\tan x-\sin x \)

Solution:

\[ \cosec x(\sec x-1)-\cot x(1-\cos x) \]

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

\[ =\frac{1-\cos x}{\sin x\cos x} -\frac{\cos x(1-\cos x)}{\sin x} \]

\[ =\frac{1-\cos x-\cos^2 x(1-\cos x)}{\sin x\cos x} \]

\[ =\frac{(1-\cos x)(1-\cos^2 x)}{\sin x\cos x} \]

\[ =\frac{(1-\cos x)\sin^2 x}{\sin x\cos x} \]

\[ =\frac{\sin x(1-\cos x)}{\cos x} \]

\[ =\frac{\sin x}{\cos x}-\sin x \]

\[ =\tan x-\sin x \]

Hence proved.