Question

Prove the following with Euler's formula to write e^itheta and e^-itheta in terms of sine and cosine. Then subtract them.

Answer 1

Euler's formula tells us that


`e^(i\theta)=\cos\theta+i\sin\theta`

`e^(-i\theta)=\cos\theta-i\sin\theta`

Suppose we subtract the two. This eliminates the cosine terms.


`e^(i\theta)-e^(-i\theta)=i\sin\theta-(-i\sin\theta)=2i\sin\theta`

Divide both sides by
`2i` and you're done.