trigonometry - Verifying the trigonometric identity $cos{x} - frac{cos{x}}{1 - tan{x}} = frac{sin{x} cos{x}}{sin{x} - cos{x}}$

I have the following trigonometric identity

$$\cos{x} - \frac{\cos{x}}{1 - \tan{x}} = \frac{\sin{x} \cos{x}}{\sin{x} - \cos{x}}$$

I've been trying to verify it for almost 20 minutes but coming up with nothing

Thank you

Answer

Observe that we need to eliminate $\tan x$

So, using $\tan x=\frac{\sin x}{\cos x},$

$$\cos x-\frac{\cos x}{1-\tan x}$$

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

$$=\cos x\left(1-\frac{\cos x}{\cos x-\sin x}\right) (\text{ multiplying numerator & denominator by }\cos x)$$

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

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