Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

How would I prove the following trig identity?

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

I am not sure where to begin any help would be useful.

share|cite|improve this question
Hint: $\cos x = \cos \left(\frac x2 + \frac x2\right)$ ... and you now a formula for $\cos(a+b)$, don't you? – martini Jul 28 '12 at 15:45
Another hint: $\cos(x/2) = \frac{1}{2}(e^{ix/2}+e^{-ix/2})$. – marlu Jul 28 '12 at 15:46
I have never seen the second one marlu commented but I know cos(a+b) is cosAcosB-cosAsinB. – Fernando Martinez Jul 28 '12 at 15:47
up vote 3 down vote accepted

I am sure you know the formula $\cos(a+b) = \cos a \cos b - \sin a \sin b$. Let $a=b = \frac{x}{2}$, which gives $\cos x = (\cos \frac{x}{2})^2 - (\sin \frac{x}{2})^2$. Since $(\cos \frac{x}{2})^2 + (\sin \frac{x}{2})^2 = 1$, this gives $\cos x = (\cos \frac{x}{2})^2 + (\cos \frac{x}{2})^2 -1$, which is your formula above.

The other follows a similar approach, except you replace the $(\cos \frac{x}{2})^2$ term instead of the $(\sin \frac{x}{2})^2$ term.

Here is the second part explicitly:

We already have $\cos x = (\cos \frac{x}{2})^2 - (\sin \frac{x}{2})^2$. Since $(\cos \frac{x}{2})^2 + (\sin \frac{x}{2})^2 = 1$, this gives $(\cos \frac{x}{2})^2 = 1-(\sin \frac{x}{2})^2$. Substituting gives $\cos x = 1-(\sin \frac{x}{2})^2 - (\sin \frac{x}{2})^2 = 1-2 (\sin \frac{x}{2})^2$.

share|cite|improve this answer
For my second one I have sin(x/2)cos(x/2)+cos(x/2)sin(x/2) but how would I proceed do I need to add them or something? – Fernando Martinez Jul 28 '12 at 16:18
Just replace $(\cos \frac{x}{2})^2$ by $1-(\sin \frac{x}{2})^2$. – copper.hat Jul 28 '12 at 16:19
Your question doesn't have a $\sin x$ expansion? – copper.hat Jul 28 '12 at 16:21
Now I have sin(x/2)(1-sin(x/2)+(1-sinx/2)sin(x/2) – Fernando Martinez Jul 28 '12 at 16:24
Still I am unsure what to.... – Fernando Martinez Jul 28 '12 at 16:26

Paint is here to help you! Just stare at the image below long enough and realize everything is ok. After, use Pythagoras' theorem:

$$ \begin{aligned} (1 + \cos(x))^2 + \sin^2(x) \\ = (2\cos(x/2))^2 \\ = 1 + 2\cos(x) + (\cos^2(x) + \sin^2(x))\\ = 2 + 2\cos(x) \\ = 4\cos^2(x/2). \end{aligned}$$

Now, you can write

$$1+\cos(x) = 2\cos^2(x/2)\\\\ \Longrightarrow \cos(x) = 2\cos^2(x/2) - 1.$$


share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.