I have to use Euler's Formula to prove that:

$$\cos^2(\theta) = \frac{\cos(2\theta)+1}{2}.$$

I have managed to prove this using trigonometric identities but I'm not sure how to use Euler's Formula or how it links into the question.

My method so far has been:

$$\frac{(\cos(2\theta)+1)}{2} = \frac{(\cos^2(\theta) - \sin^2(\theta)+1)}{2}$$



So $$\frac{(\cos(2\theta)+1)}{2} =\frac{2\cos^2(\theta)}{2} =\cos^2(\theta).$$


Eulers identity $e^{i\theta} = \cos \theta + i\sin\theta$

$e^{i\theta} + e^{-i\theta} = 2\cos \theta\\ \frac 14 (e^{i\theta} + e^{-i\theta})^2 = \cos^2 \theta\\ \frac 14 (e^{2i\theta} + e^{-2i\theta} + 2) = \cos^2 \theta\\ \frac 14 (2\cos 2\theta + 2) = \cos^2 \theta\\ \frac 12 (\cos 2\theta + 1) = \cos^2 \theta$


By Euler's formula $2\cos x=e^{ix}+e^{-ix}$

$$2\cos2\theta=(e^{i\theta}+e^{-i\theta})^2-2e^{i\theta}\cdot e^{-i\theta} =(2\cos\theta)^2-2$$


Euler's formula can be used to prove the addition formula for both sines and cosines as well as the double angle formula (for the addition formula, consider $\mathrm{e^{ix}}$. The same method as the double angle formula works).

From Euler's formula, $\mathrm{e^{2ix}=cos(2x)+isin(2x)=(e^{ix})(e^{ix})=(cos(x)+isin(x))(cos(x)+isin(x))}$

Expand the brackets and compare the real and imaginary parts for the double angle formulas.


$$\cos x = \frac{e^{ix} +e^{-ix}}{2}$$ Assume $e^{ix}=u$ $$\left(\cos x\right)^2 = \left(\frac{u +\frac{1}{u}}{2}\right)^2$$ $$\cos^2 x=\frac{u^2 +\frac{1}{u^2}}{4} +\frac{2}{4}$$ $$\cos^2 x = \frac{e^{2ix} +e^{-2ix}}{4 }+ \frac{1}{2}$$


Start with Euler's formula:


Squaring both sides gives:


Which, using the laws of exponents and the expansion of brackets, becomes:


The left can be written with the exponent as a multiple of $i$ and the right can be simplified because $i^{2}=-1$:


However as $e^{i(2\theta)}$ is in the form $e^{ix}$ it can be expressed through Euler's formula:


The real parts can be equated:


Which leaves:


As $\sin^2{\theta}=1-\cos^2{\theta}$:


This bracket can be multiplied out to give:


Which can be rearranged to:



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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