Prove that e^{-ix}=\cos(x)-i\sin(x)

Agohofidov6

Agohofidov6

Answered question

2021-12-11

Prove that eix=cos(x)isin(x)

Answer & Explanation

Pansdorfp6

Pansdorfp6

Beginner2021-12-12Added 27 answers

sin(x)=xx33!+x55!+
cos(x)=1x22!+x44!+
ex=1+x+x22!+x33!+xP{4}4!+
Multiply ex by i
1+(ix)+(ix)22!+(ix)33!+(ix)44!+
Expand
1ixx22!+x33!+x44!+
Factorise it:
(1x22!+x44!+)i(xx33!+x55!+)
And the first part of the equation is equal to cos(x) and the second part to sin(x). Thus, we can replace them.
(cos(x))i(sin(x))
cos(x)isin(x)
Annie Gonzalez

Annie Gonzalez

Beginner2021-12-13Added 41 answers

You can prove this using Taylor's Series.

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