Question 515391
Recall that {{{csc x = (1)/(sin x)}}}.  So we substitute this into our expression and work from there:

{{{ (cos x)/(csc x - sin x) }}}
{{{ (cos x)/((1)/(sin x) - sin x))}}} (substituting)
{{{ (cos x)/((1)/(sin x) - (sin x)^2/(sin x))}}} (putting all of the terms in the denominator under a common denominator)
{{{ (cos x)/((1 - (sin x)^2)/(sin x))}}} (combining fractions in the denominator)
{{{ (cos x) * ((sin x)/(1 - (sin x)^2))}}} (simplifying the complex fraction by multiplying by the reciprocal of the denominator)

Now recall that {{{(cos x)^2 + (sin x)^2 = 1}}}, so {{{(cos x)^2 = 1 - (sin x)^2}}}.  So, substituting {{{(cos x)^2}}} for {{{1 - (sin x)^2}}}, we get:

{{{ (cos x) * ((sin x)/((cos x)^2))}}} (substituting)
{{{ 1 * ((sin x)/(cos x))}}} (canceling one {{{cos x}}})
{{{ tan x }}} (remembering that {{{tan x = (sin x)/(cos x)}}})

So our expression simplified to {{{tan x}}}.