"I’ve got to proof that:tan⁡(A2)=1−cos⁡(A)1+cos⁡(A)" was answered by students like you

Bruce Partridge

Answered question

2022-02-26

I’ve got to proof that:

\tan (\frac{A}{2}) = \sqrt{\frac{1 - \cos (A)}{1 + \cos (A)}}

Answer & Explanation

zahrkao8vm

Beginner2022-02-27Added 10 answers

You can solve most trignometric identities using Euler's formula:
$e^{i θ} = \cos (θ) + i \sin (θ)$
Solving for sin and cos, we see:
$\cos (θ) = \frac{1}{2} (e^{- i θ} + e^{i θ})$
$\sin (θ) = \frac{1}{2} i (e^{- i θ} - e^{i θ})$
Plugging in the formula above on the right side yields:
$\sqrt{\frac{1 + \frac{1}{2} (- e^{- i θ} - e^{i θ})}{1 + \frac{1}{2} (e^{- θ} + e^{i θ})}}$
Algebraic simplication yields:
$\sqrt{- \frac{(- 1 + e^{i θ)^{2}}}{(1 + e^{i θ})^{2}}}$
Now, replacing the left side with sin/cos and substituting as above yields:
$\frac{i (e^{- \frac{i θ}{2}} - e^{\frac{i θ}{2}})}{e^{- \frac{i θ}{2}} + e^{\frac{i θ}{2}}}$
Simplifying yields:
$- \frac{i (- 1 + e^{i θ}}}{1 + e^{i θ}}$
which is identical to the right side after you take the square root.

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