Show that:Cosx-cos^3 x/sin=sin x cos x. ?

To prove that:

(cosx - cos^3 x) / sin x = sin x cos x

We will start by simplifying the left side of the equation:

(cosx - cos^3 x) / sin x
= cosx(1 - cos^2 x) / sin x
= cosx sin^2 x / sin x
= cosx sin x
= sin x cos x

Therefore, we have shown that (cosx - cos^3 x) / sin x = sin x cos x.

$$L.H.S = \cos x - \frac{{\cos }^3 x}{\sin x}$$
$$= \cos x - \frac{{\cos }^2 x \cos x}{\sin x}$$
$$= \cos x - \frac{{\cos }^2 x}{\tan x}$$
$$= \cos x - \cos x = 0$$

$$R.H.S = \sin x \cos x$$
$$\therefore L.H.S=R.H.S$$