Euler's formula, also known as Euler's identity, states that e^(itheta) = cos(theta) + isin(theta) for all real numbers theta.
To prove this formula, we can start with the Taylor series expansions of e^(ix) and cos(x) + i*sin(x).
e^(ix) = 1 + ix + (ix^2)/2! + (ix^3)/3! + ...
cos(x) + i*sin(x) = 1 - (x^2)/2! + (ix^2)/2! - (x^4)/4! + (ix^4)/4! - ...
Matching the real and imaginary parts of these two equations, we see that:
1 = 1
ix = ix
(ix^2)/2! = -(x^2)/2!
(ix^3)/3! = (x^3)/3!
...
Since the coefficients of each term match on both sides of the equation, we can conclude that e^(ix) = cos(x) + i*sin(x) for all real numbers x, which is exactly the Euler's formula.
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