Precalculus, Chapter 5, 5.2, Section 5.2, Problem 30

Verify the identity cos(theta)cot(theta)/(1-sin(theta))-1=csc(theta)
(cos(theta)cot(theta))/(1-sin(theta))-1=csc(theta)
Rewrite cot(theta) as a quotient.
[[cos(theta)/1]*[cos(theta)/sin(theta)]]/(1-sin(theta))-1=csc(theta)

[(cos^2(theta))/sin(theta)]/[(1-sin(theta)]]-[1]=csc(theta)

[(cos^2(theta))/sin(theta)]*1/(1-sin(theta))-1=csc(theta)
Use the pythagorean identity sin^2(theta)+cos^2(theta)=1 to substitute in for the cos^2(theta).
[1-sin^2(theta)]/sin(theta)*[1/(1-sin(theta))]-1=csc(theta)
Factor the term 1-sin^2(theta).
[(1+sin(theta)(1-sin(theta))]/sin(theta)]*[1/(1-sin(theta))]-1=csc(theta)
Cancel the (1-sin(theta)).
[(1+sin(theta))/sin(theta)]*[1]-1=csc(theta)
Rewrite the first term as two separate fractions.
1/sin(theta)+sin(theta)/sin(theta)-1=csc(theta)

csc(theta)+1-1=csc(theta)

csc(theta)=csc(theta)