# Problem with sin(x/2) proof

1. Oct 13, 2011

### Uniquebum

My problem is:
Proof $sin(\frac{x}{2}) = \pm \sqrt{\frac{1-cos(x)}{2}}$

Simple issue really i'd think but i can't come up with a way.

For starters i'd use however
$cos^2(x) + sin^2(x) = 1$ identity.

$sin(\frac{x}{2}) = \pm \sqrt{1-cos^2(\frac{x}{2})}$

But then i got nothing...

2. Oct 13, 2011

### Mentallic

Do you know the expansion for cos(2x) in terms of sin(x) and cos(x)? From there you would convert this expression solely into terms with sin(x), and finally solve for sin(x).

3. Oct 13, 2011

### Uniquebum

Ahhh i get it!
$cos(2\frac{x}{2}) = cos^2(\frac{x}{2})-sin^2(\frac{x}{2})$
$cos^2(\frac{x}{2}) = cos(2\frac{x}{2})-sin^2(\frac{x}{2})$

Thus
$sin^2(\frac{x}{2}) = 1-cos^2(\frac{x}{2})$
$sin^2(\frac{x}{2}) = 1-cos(x)-sin^2(\frac{x}{2})$

And so
$sin(\frac{x}{2}) = \pm \sqrt{\frac{1-cos(x)}{2}}$
Thanks!

4. Oct 13, 2011

### Staff: Mentor

I'm not sure you do.
No,
$cos^2(\frac{x}{2}) = cos(2\frac{x}{2}) + sin^2(\frac{x}{2})$

5. Oct 13, 2011

### Uniquebum

That was a typo but anyway... :)

6. Oct 13, 2011

Nice work