SOLUTION: Would be ever so much grateful for help to prove the trigonometric relation in a thoroughly detailed explanation of proof. {{{tan (x/2)= (1-cosx)/ (sinx)}}}

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Question 667155: Would be ever so much grateful for help to prove the trigonometric relation in a thoroughly detailed explanation of proof.
tan+%28x%2F2%29=+%281-cosx%29%2F+%28sinx%29
Found 2 solutions by tam144, Edwin McCravy:
Answer by tam144(6) About Me  (Show Source):
You can put this solution on YOUR website!
Start from the right side of the equation. Refer to List of Trig Identities
in book titled:"Solving trig equations and inequalities" (Amazon e-book 2010)
1 - cos x = 2 sin^2 x/2 (Identity 11B)
sin x = 2sin x/2.cos x/2 (Identity 10)
Quotient: (1 - cos x)/sin x = (2sin^2 x/2)/(2sin x/2.cos/x/2)
After simplification, the quotient becomes:
!1 - cos x)/sin x = (sin x/2) / (cos x/2) = tg x/2

Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!

tan+%28x%2F2%29=+%281-cosx%29%2F+%28sinx%29

Let x%2F2 = theta
which means x = 2theta

Then the identity becomes

tan%28theta%29 = %281-cos%282theta%29%29%2F+%28sin%282theta%29%29

We will work with the right side:
Use the identities cos%282theta%29+=+cos%5E2%28theta%29+-+sin%5E2%28theta%29 and sin%282theta%29+=+2sin%28theta%29cos%28theta%29

      = %281+-+%28cos%5E2%28theta%29-sin%5E2%28theta%29%29%29%2F%282sin%28theta%29cos%28theta%29%29

      = %281+-+cos%5E2%28theta%29%2Bsin%5E2%28theta%29%29%2F%282sin%28theta%29cos%28theta%29%29
  
Use the identity 1-cos%5E2%28theta%29+=sin%5E2%28theta%29 in the numerator

      = %28sin%5E2%28theta%29%2Bsin%5E2%28theta%29%29%2F%282sin%28theta%29cos%28theta%29%29

      = %282sin%5E2%28theta%29%29%2F%282sin%28theta%29cos%28theta%29%29

      = 

      = sin%28theta%29%2Fcos%28theta%29

Use the identity sin%28theta%29%2Fcos%28theta%29+=+tan%28theta%29

      = tan%28theta%29

Substitute x%2F2 for theta

      = tan%28x%2F2%29

Edwin