Expert: Sherman D. - 2/9/2008

Question
(1-cos)/(1+cos)= (csc-cot)^2

Answer
(1 - cos(x))/(1 + cos(x)) = (csc(x) - cot(x))^2

(csc(x) - cot(x))^2 =
(csc(x) - cot(x))(csc(x) - cot(x)) =
csc(x)^2 - csc(x)cot(x) - csc(x)cot(x) + cot(x)^2 =
csc(x)^2 + cot(x)^2 - 2csc(x)cot(x)
(1/sin(x))^2 + (cos(x)/sin(x))^2 - 2(1/sin(x))(cos(x)/sin(x))
((1 + cos(x)^2)/sin(x)^2) - (2cos(x)/sin(x)^2)
(1 + cos(x)^2 - 2cos(x))/(sin(x)^2)
(cos(x)^2 - 2cos(x) + 1)/(sin(x)^2)
((cos(x) - 1)(cos(x) - 1))/(1 - cos(x)^2)
((-1 + cos(x))(-1 + cos(x)))/((1 - cos(x))(1 + cos(x)))
((1 - cos(x))^2 / ((1 - cos(x))(1 + cos(x))

(1 - cos(x))/(1 + cos(x))

so

(1 - cos(x))/(1 + cos(x)) = (csc(x) - cot(x))^2