How do I simplify this problem (so that the Right Side is equal to the Left Side)?
cosx - (cosx)/(1-tanx) = (sinx cosx)/ (sinx - cosx)
You are right so far except your denominator has disappeared.
It must be:
$\displaystyle \frac{cos x-\frac{sin x cos x}{cosx}-cos x}{1-tan x}$
The numerator can be simplified to
$\displaystyle \frac{cos x-sinx-cos x}{1-tan x}= \frac{-sinx}{1-tan x}$
Then as previously stated change $\displaystyle tan x$ in the denominator to $\displaystyle \frac{sinx}{cos x}$
$\displaystyle -\frac{sin x}{1-\frac{sin x}{cosx}}$
$\displaystyle -\frac{sin x}{\frac{cos x}{cos x}-\frac{sin x}{cos x}}$ Common denominators
$\displaystyle -\frac{sin x}{\frac{cos x - sin x}{cos x}}$
$\displaystyle -sin x \cdot \frac{cos x}{cos x - sin x}$ Multiply the reciprocal
$\displaystyle -\frac{sin x \cdot cos x}{-(sin x - cos x)}$ Simplify the bottom
$\displaystyle \frac{sin x \cdot cos x}{(sin x - cos x)}=RHS$