Precalculus, Chapter 5, 5.2, Section 5.2, Problem 32

Verify the identity. cos(x)-[cos(x)/(1-tan(x))]=(sinxcosx)/(sinx-cosx)
[cos(x)(1-tan(x))-cos(x)]/[1-tan(x)]=[sin(x)cos(x)]/[sin(x)-cos(x)]
[cos(x)-tan(x)cos(x)-cos(x)]/(1-tan(x))=[sin(x)cos(x)]/[sin(x)-cos(x)]
[-tan(x)cos(x)]/[1-tan(x)]=[sin(x)cos(x)]/[sin(x)-cos(x)]
Rewrite the tan(x) terms as the quotient sin(x)/cos(x).
-[[sin(x)/cos(x)]cos(x)]/[1-(sin(x)/cos(x))]=(sin(x)cos(x))/(sin(x)-cos(x))
-sin(x)/[(cos(x)-sin(x))/cos(x)]=(sin(x)cos(x))/(sin(x)-cos(x))
-[sin(x)/1]*[cos(x)/(cos(x)-sin(x))]=(sin(x)cos(x))/(sin(x)-cos(x))
(sin(x)cos(x))/-(cos(x)-sin(x))=(sin(x)cos(x))/[sin(x)-cos(x)]
(sin(x)cos(x))/[-cos(x)+sin(x)]=(sin(x)cos(x))/(sin(x)-cos(x))
(sin(x)cos(x))/(sin(x)-cos(x))=(sin(x)cos(x))/(sin(x)-cos(x))