Precalculus, Chapter 5, 5.2, Section 5.2, Problem 26

Transform the left side:
sec(x)*(csc(x)-2sin(x))=(1/cos(x))*((1-2sin^2(x))/sin(x)=(cos^2(x)-sin^2(x))/(cos(x)*sin(x)).
The right side is
cos(x)/sin(x)-sin(x)/cos(x)=(cos^2(x)-sin^2(x))/(sin(x)*cos(x)),
which is the same.