I know that using Euler's Formula we can write cosine like the first expression, but concerning the second expression, is it correct like that?
$$\cos(\theta)=\frac{e^{i\theta}+e^{-i\theta}}2$$
$$\cos(2\theta)=\frac{2e^{i\theta}+2e^{-i\theta}}2$$
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4The operation you are performing is called "substitution", and it means to systematically replace each instance of the substitution variable (in this case, $\color{#00F}\theta$) with the desired replacement (in this case, $\color{#C00}{2\theta}.)\quad$ So, $\displaystyle\cos(\color{#C00}{2\theta})=\frac{e^{i(\color{#C00}{2\theta})}+e^{-i(\color{#C00}{2\theta})}}2.$ – ryang Apr 06 '21 at 15:27
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2@RyanG nice use of colours! – A-Level Student Apr 06 '21 at 21:19
3 Answers
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No, this is not correct. You have to substitute $2\theta$ for $\theta$ which means that $$ \cos(2\theta) = \frac{e^{2i\theta} + e^{-2i\theta}}{2}. $$
Martin Vesely
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No, it is not correct. Since$$\cos\theta=\frac{e^{i\theta}+e^{-i\theta}}2,$$you have$$\cos(2\theta)=\frac{e^{2i\theta}+e^{-2i\theta}}2.$$
José Carlos Santos
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Expression 1 $\to \cos 0=1$, Expression 2 $\to \cos 0=2>1$.
Can the second expression be correct?
Star Bright
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