sinx/(1 - 1/cosx) - sinx/(1 + 1/cosx)
sinx/((cosx - 1)/cosx) - sinx/((cosx + 1)/cosx)
sinx xx cosx/(cosx - 1) - sinx xx cosx/(cosx + 1)
(sinxcosx)/(cosx - 1) - (sinxcosx)/(cosx + 1)
((sinxcosx)(cosx + 1))/((cosx - 1)(cosx + 1)) - ((sinxcosx)(cosx- 1))/((cosx + 1)(cosx - 1))
(sinxcos^2x + sinxcosx)/(cos^2x - 1) - (sinxcos^2x - sinxcosx)/(cos^2x - 1)
(sinxcos^2x + sinxcosx - sinxcos^2x + sinxcosx)/(cos^2x - 1)
(2sinxcosx)/(cos^2x - 1)
Applying the double angle identity 2sinxcosx = sin2x and the pythagorean identity cos^2x - 1 = -sin^2x we get the following:
(sin2x)/(-sin^2x) -> this is your answer.
*Beware: This is not the only possible answer; there are many ways of attacking this problem, with different answers completely within the realm of possibility. *
Hopefully this helps!
