Question

Prove that : e^ix = Cosx + i.Sin x​

Answer 1

Answer:

Refer the attachment!!:)

Answer 2

Can you prove the below equality? eix=cos(x)+isin(x)

This is how Leonard Euler did it.

Paging through a wonderful book “An imaginary tale: The story of −1−−−√ by Paul J. Nahin (strongly recommend!), I discovered this episode of history.

On 18 October 1740 Euler wrote to John Bernoulli that the solution to differential equation of a harmonic oscillator

y”+y=0,y(0)=2,y'(0)=0

can be written in two ways:

y(x)=2cos x

and

y(x)=eix+e−ix.

He concluded from that

2cosx=eix+e−ix.

which was first step to his famous formula. After differentiating the last fomula with respect to x, one gets

−2sinx=ieix−ie−ix,

or

2isinx= eix−e−ix.

Adding expressions for 2cosx and 2 i sinx, one concludes

2cosx+2isinx=2eix,

that is, the desired formula

eix=cosx+isinx.

Obviously, Euler was using the uniqueness of a solution with given initial values. I bet his belief in the uniqueness was rooted in physical intuition. IMHO, for Euler, expansion of mathematical language did not change his vision of the world. I would not be surprised if he was thinking that an “imaginary” solution corresponded to something in the real world, something that was not discovered yet.

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