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Prove e^(ix) = cos (x) + isin(x)

We first write each side of the equation using the maclaurin series for each function.

eix = 1 + ix + (ix)2/2! + (ix)3/3! + (ix)4/4! + ......

eix = 1 + ix - x2/2! - ix3/3! + x4/4! + .....

cos(x) + isin(x) = (1 - x2/2! + x4/4! - x6/6! +....) + i(x - x3/3! + x5/5! - x7/7! + ......)

writing the above equation in increasing powers of x:

cos(x) + isin(x) = 1 + ix - x2/2! - ix3/3! + x4/4! + ....

As seen the maclaurin series for each side of the equation are the same hence eix = cos(x) + isin(x)

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