Now remember, e to the i theta was our shorthand notation for

cosine theta plus i sine theta.

Therefore either -i theta or you just replaced theta with minus theta,

it's cosine of minus theta plus i times the sine of minus theta, but

cosine is a need function.

These are just our regular viewed value of cosine and sine functions.

So that's cosine of theta but sine as an odd function, so

sine of -theta is -sine of theta.

Now look at these two identities, e to the i theta is cosine theta + i

sine theta, e to the- i theta is cosine theta- i sine theta.

If I add these two things up, What I

find is that e to the i theta + e to the- i theta is simply two cosine thetas.

The i sine theta part cancels out because it has a plus sign here and

a minus sign there.

On the other hand, if I subtract these two equations from each other,

I find that e to i theta- e to the -i theta is 2i sine theta,

because now the cosines cancel out and the sines don't.

But these two equations that I just obtained, I can solve them for

cosine theta and sine theta by simply dividing by 2 or 2i.