How to Derive Euler's Identity?

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To derive cos(θ) = 1/2 (e^{iθ} + e^{-iθ}) from e^{iθ} = cos(θ) + isin(θ), one can add e^{iθ} and e^{-iθ}, which cancels out the imaginary parts, resulting in the expression for cosine. Similarly, for sin(θ) = 1/2i (e^{iθ} - e^{-iθ}), subtracting e^{-iθ} from e^{iθ} isolates the imaginary component, yielding the sine function. The factor of 1/2 arises from the need to average the two exponential terms. The discussion also references a related thread that helped clarify the derivation. Overall, the key steps involve manipulating the exponential forms of sine and cosine.
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Hello. Please tell me how do I derive:

cos(θ)=1/2 (e^{iθ}+e^{-iθ})

from:

e^{iθ}=cos(θ) + isin(θ)

as well as:

sin(θ)=1/2i (e^{iθ}-e^{-iθ})

I can't figure it out...for example, where does the 1/2 come from? Thank you:smile:
 
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Write out what ##e^{-i\theta}## is.

Edit: ninja'd by the OP
 

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