Precalculus, Chapter 5, 5.2, Section 5.2, Problem 31

Verify the identity: 1/(cos(x)+1)+1/(cos(x)-1)=-2csc(x)cot(x)
[cos(x)-1+cos(x)+1]/(cos^2(x)-1)=-2csc(x)cot(x)
[2cos(x)]/(cos^2(x)-1)=-2csc(x)cot(x)
Use the pythagorean identity sin^2(x)+cos^2(x)=1 to simplify the denominator.
If cos^2(x)-1 is isolated the equation would be cos^2(x)-1=-sin^2(x).

[2cos(x)]/[-sin^2(x)]=-2csc(x)cot(x)
[-2cos(x)]/[sin(x)sin(x)]=-2csc(x)cot(x)
-2*[1/sin(x)]*[cos(x)/sin(x)]=-2csc(x)cot(x)
Use the reciprocal identity csc(x)=1/sin(x). Also use the quotient identity cot(x)=cos(x)/sin(x).
-2csc(x)cot(x)=-2csc(x)cot(x)