# Proving a Trigonometric Identity

• Dec 20th 2009, 06:51 PM
KelvinScale
Proving a Trigonometric Identity
Hi everyone:

Is there any way to prove the following identity:

tan(A/2)=(sinA)/(1+cosA)

without drawing a diagram? I know that it is a basic trig identity, but I don't know how to prove it by manipulating hte formulas.

Thanks!
• Dec 20th 2009, 07:30 PM
bigwave
use sin(x/2) / cos(x/2)
$\displaystyle \tan{\frac{A}{2}}=\frac{\sin{A}}{1+\cos{A}}$

this derived from by using

$\displaystyle \tan{\frac{A}{2}}=\frac{\sin\frac{A}{2}}{\cos\frac {A}{2}}$

$\displaystyle \frac {\pm\sqrt{\frac{1-\cos{A}}{2}}} {\pm\sqrt{\frac{1+\cos{A}}{2}}}$

$\displaystyle =\pm\sqrt\frac{\left(1-\cos{A}\right)\times\left(1+cos{A}\right)} {\left(1+\cos{A}\right)\times\left(1+\cos{A}\right )}$

$\displaystyle =\pm\sqrt\frac{1-\cos^2{A}} {\left(1+\cos{A}\right)^2}$

$\displaystyle =\pm\vert\frac{\sin{A}}{1-\cos{A}}\vert$

$\displaystyle 1-\cos{A}$ in never negative, so the sign of the fractional expression depends only on the sign of $\displaystyle \sin{A}$
• Dec 20th 2009, 07:36 PM
Soroban
Hello, Kelvin!
We need these two identities: .$\displaystyle \begin{array}{ccc}\sin\frac{\theta}{2} &=& \sqrt{\dfrac{1-\cos\theta}{2}} \\ \\[-4mm]\cos\frac{\theta}{2} &=& \sqrt{\dfrac{1+\cos\theta}{2}} \end{array}$

Quote:

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

$\displaystyle \tan\frac{A}{2} \;=\;\frac{\sin\frac{A}{2}}{\cos\frac{A}{2}} \;=\;\frac{\sqrt{\dfrac{1-\cos A}{2}}} {\sqrt{\dfrac{1 + \cos A}{2}}} \;=\; \sqrt{\frac{1-\cos A}{1+\cos A}}$

Multiply by $\displaystyle \frac{1+\cos A}{1 + \cos A}$

. . $\displaystyle \sqrt{\frac{1-\cos A}{1+\cos A}\cdot\frac{1+\cos A}{1 + \cos A}} \;=\;\sqrt{\frac{1-\cos^2\!A}{(1+\cos A)^2}} \;=\;\sqrt{\frac{\sin^2\!A}{(1+\cos A)^2}} \;=\; \frac{\sin A}{1 + \cos A}$

• Dec 20th 2009, 07:38 PM
I-Think
$\displaystyle tan(\frac{A}{2})=\frac{sinA}{1+cosA}$

Manipulating R.H.S.
$\displaystyle \frac{2sin(\frac{A}{2}){cos(\frac{A}{2})}}{1+cos^2 (\frac{A}{2})-sin^2(\frac{A}{2})}$

Note that
$\displaystyle 1=sin^2(\frac{A}{2})+cos^2(\frac{A}{2})$

Introduce into the equation and you should solve your problem

Edit
In the time I took to write this I was beaten by 2 other forum users
I really need to take typing classes.
On the other hand, at least I provided a different method
• Dec 20th 2009, 07:41 PM
KelvinScale
Thank you so much!
• Dec 20th 2009, 09:11 PM
KelvinScale
Okay, Similar Question.

Prove that:

tan(A/2)=(1+sinA-cosA)/(1+sinA+cosA)

I tried replacing tan(A/2) with sinA/(1+cosA), but I could not find a way to add the extra components to both the numerator and denominator. Gosh this is frustrating.
• Dec 21st 2009, 05:55 AM
Soroban
Helolo, Kelvin!

This one is tricky . . .

Quote:

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

Multiply by $\displaystyle \frac{1+\sin A - \cos A}{1 + \sin A - \cos A}$

. . $\displaystyle \frac{1 + \sin A - \cos A}{1 + \sin A + \cos A}\cdot{\color{blue}\frac{1 + \sin A - \cos A}{1 + \sin A - \cos A}} \;=\;\frac{(1+\sin A - \cos A)^2}{(1+\sin A)^2 - \cos^2\!A}$

. . $\displaystyle =\; \frac{1 + 2\sin A - 2\cos A + \sin^2\!A - 2\sin A\cos A + \cos^2\!A}{1 + 2\sin A + \sin^2\!A - \cos^2\!A}$

. . $\displaystyle =\; \frac{1 + \overbrace{\sin^2\!A + \cos^2\!A}^{\text{This is 1}} + 2\sin A - 2\cos A - 2\sin A\cos A}{2\sin A + \sin^2\!A + \underbrace{1 - \cos^2\!A}_{\text{This is }\sin^2\!A}}$

. . $\displaystyle =\; \frac{2 + 2\sin A - 2\cos A - 2\sin A\cos A}{2\sin A + 2\sin^2\!A} \;=\;\frac{2(1 + \sin A - \cos A - \sin A\cos A}{2\sin A(1 + \sin A)}$

. . $\displaystyle =\; \frac{1 + \sin A - \cos A - \sin A\cos A}{\sin A(1 + \sin A)} \;=\;\frac{(1 + \sin A) - \cos A(1 + \sin A)}{\sin A(1 + \sin A)}$

. . $\displaystyle =\; \frac{(1+\sin A)(1 - \cos A)}{\sin A(1 + \sin A)} \;=\;\frac{1-\cos A}{\sin A}$

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

. . . . . . . . . . . . . $\displaystyle =\;\frac{\sin^2\!A}{\sin A(1 + \cos A)} \;=\;\frac{\sin A}{1 + \cos A}$

Finally: .$\displaystyle \frac{\sin A}{1 + \cos A} \;=\;\tan\frac{A}{2} \quad \hdots\;There!$

• Dec 21st 2009, 07:00 AM
Krizalid
be careful with what angle you're working on, those identities are not true for all angles.