How can i prove that cot x/2 = (sinx)/(1-cosx)?

Question

i applied cosx/sinx but i ended up with
(1+cosx)/(1-cosx)
thanks

Answer 1

1 = sin²(x/2)+cos²(x/2)
1–cos²(x/2) = sin²(x/2)
1–cos²(x/2)+sin²(x/2) = 2*sin²(x/2)
1–cos(x) = 2*sin²(x/2)
cos(x/2)*[1–cos(x)] = 2*sin²(x/2)*cos(x/2)
cos(x/2)*[1–cos(x)] = sin(x)*sin(x/2)
[cos(x/2)]/[sin(x/2)] = sin(x)/[1–cos(x)]
cot(x/2) = sin(x)/[1–cos(x)]
Q.E.D.

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Answer 2

(a million + sinx + cosx)(a million - cosx) = a million + sinx + cosx - cosx - cosxsinx - cos²x. = sinx + sin²x - cosxsinx, considering the fact that sin²x + cos²x = a million. sinx(a million + sinx - cosx) = sinx + sin²x - sinxcosx. hence (a million + sinx + cosx)(a million - cosx) = sinx(a million + sinx - cosx). Now divide the two factors via (a million + sinx - cosx)(a million - cosx) to reach on the needed identity.

Answer 3

If you're flustered, it might be easier to try to prove cot y = sin 2y/(1-cos2y)

They're exactly the same thing, after all. :)

And by the way, cos2y = cos^2 - sin^2 = 1 - 2sin^2.

From there, it's pretty easy.