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

Solution:

\[ (\cosec x-\sin x)(\sec x-\cos x)(\tan x+\cot x) \]

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

\[ =\frac{1-\sin^2 x}{\sin x}\cdot \frac{1-\cos^2 x}{\cos x}\cdot \frac{\sin^2 x+\cos^2 x}{\sin x\cos x} \]

\[ =\frac{\cos^2 x}{\sin x}\cdot \frac{\sin^2 x}{\cos x}\cdot \frac{1}{\sin x\cos x} \]

\[ =\frac{\cos^2 x\sin^2 x}{\sin^2 x\cos^2 x} \]

\[ =1 \]

Hence proved.