# Math Help - Identity

1. ## Identity

sinA / (1+cosA) = tan 0.5A

2. Hello, Punch!

We need a few identities: . $\begin{array}{cccccccccc}\sin^2\!\frac{A}{2} &=& \dfrac{1-\cos A}{2} && \Rightarrow && \sin\frac{A}{2} &=& \sqrt{\dfrac{1-\cos A}{2}} \\ \\[-3mm]
\cos^2\!\frac{A}{2} &=& \dfrac{1+\cos A}{2} && \Rightarrow && \cos\frac{A}{2} &=& \dfrac{1+\cos A}{2}\end{array}$

Prove: . $\frac{\sin A}{1+\cos A}\: =\: \tan \tfrac{A}{2}$

The right side is: . $\tan\frac{A}{2} \;=\;\frac{\sin\frac{A}{2}}{\cos\frac{A}{2}} \;=\;\frac{\sqrt{\dfrac{1-\cos A}{2}}}{\sqrt{\dfrac{1+\cos A}{2}}} \;=\;\sqrt{\frac{1-\cos A}{1+\cos A}}$

Multiply by $\frac{1+\cos A}{1 + \cos A}:\quad\sqrt{\frac{1-\cos A}{1+\cos A}\cdot\frac{1+\cos A}{1+\cos A}} \;=\;\sqrt{\frac{1-\cos^2\!A}{(1+\cos A)^2}} \;=\;\sqrt{\frac{\sin^2\!A}{(1+\cos A)^2}}$

Therefore: . $\tan\frac{A}{2} \;=\;\frac{\sin A}{1 + \cos A}$

3. Originally Posted by Soroban
Hello, Punch!

We need a few identities: . $\begin{array}{cccccccccc}\sin^2\!\frac{A}{2} &=& \dfrac{1-\cos A}{2} && \Rightarrow && \sin\frac{A}{2} &=& \sqrt{\dfrac{1-\cos A}{2}} \\ \\[-3mm]$ $
\cos^2\!\frac{A}{2} &=& \dfrac{1+\cos A}{2} && \Rightarrow && \cos\frac{A}{2} &=& \dfrac{1+\cos A}{2}\end{array}" alt="
\cos^2\!\frac{A}{2} &=& \dfrac{1+\cos A}{2} && \Rightarrow && \cos\frac{A}{2} &=& \dfrac{1+\cos A}{2}\end{array}" />

The right side is: . $\tan\frac{A}{2} \;=\;\frac{\sin\frac{A}{2}}{\cos\frac{A}{2}} \;=\;\frac{\sqrt{\dfrac{1-\cos A}{2}}}{\sqrt{\dfrac{1+\cos A}{2}}} \;=\;\sqrt{\frac{1-\cos A}{1+\cos A}}$

Multiply by $\frac{1+\cos A}{1 + \cos A}:\quad\sqrt{\frac{1-\cos A}{1+\cos A}\cdot\frac{1+\cos A}{1+\cos A}} \;=\;\sqrt{\frac{1-\cos^2\!A}{(1+\cos A)^2}} \;=\;\sqrt{\frac{\sin^2\!A}{(1+\cos A)^2}}$

Therefore: . $\tan\frac{A}{2} \;=\;\frac{\sin A}{1 + \cos A}$
Hello! Thanks a lot thank you very much!

4. lol Soroban, guess you can explain it better than me.

http://www.mathhelpforum.com/math-he...-identity.html

5. Just a question, do we need the memorize all the formulas, or the formulas are given in the formula sheet during the examination?

6. Originally Posted by Punch
Just a question, do we need the memorize all the formulas, or the formulas are given in the formula sheet during the examination?
It up to the teacher. He or she may let you use a formula sheet or you may have to memorize them. You could also learn to derive them.