Thread: prove the trig identity

1. prove the trig identity

Hi, the question is: starting from the definition of the complex cosine function, prove the trig identity:
$\displaystyle cos^2z=\frac{1}{2}(1 + cos 2z)$

here's what i got
$\displaystyle cos^2 z = [\frac{1}{2}(e^{iz}+e^{-iz}]^2$
$\displaystyle = \frac{1}{4}(e^{2iz}+e^{-2iz} + 2)$
$\displaystyle = \frac{1}{2}(\frac{1}{2} (e^{2iz} + e^{-2iz} + 1)$
and now i don't know what to do.. is this right so far? and can someone please help me finish it off?

2. Originally Posted by Dgphru
Hi, the question is: starting from the definition of the complex cosine function, prove the trig identity:
$\displaystyle cos^2z=\frac{1}{2}(1 + cos 2z)$

here's what i got
$\displaystyle cos^2 z = [\frac{1}{2}(e^{iz}+e^{-iz}]^2$
$\displaystyle = \frac{1}{4}(e^{2iz}+e^{-2iz} + 2)$
$\displaystyle = \frac{1}{2}(\frac{1}{2} (e^{2iz} + e^{-2iz} + 1)$
and now i don't know what to do.. is this right so far? and can someone please help me finish it off?

You alredy did everything!

$\displaystyle \frac{1}{4}\left(e^{2iz}+e^{-2iz} + 2\right)$ $\displaystyle =\frac{1}{2}\left(\frac{e^{2iz}+e^{-2iz}}{2}+1\right)=\frac{1}{2}\left(\cos 2z+1\right)$ ...

Tonio