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Derive the formulae for sine x and cosine x in terms of exp(ix) and exp(–ix), that follows from  equation (A) above.

Solution:

Solution: Equation (A) reads:

exp(ix)  =  cos x + i sin x.

and it holds for all x positive or negative, so that we also have:

exp(-ix) =  cos x - i sin x.

where we have used the fact that cosine is an even function and sine an odd one.

We can solve these two equations for cos x and sin x: