Proof of double angle formulas using Euler's equation

[SamRoss]

Hi all,

I'm slowly working through "Mathematical Methods in the Physical Sciences" by Mary Boas, which I highly recommend, and I'm stumped on one of the questions. The problem is to prove the double angle formulas sin (2Θ)=2sinΘcosΘ and cos(2Θ)=cos²Θ-sin²Θ by using Euler's formula (raised to the second power) e^i2Θ=cos(2Θ)+isin(2Θ). I should be splitting this up into two equations, one setting the real parts equal to each other and the other with the imaginary parts but I'm not sure how to do that with the left side looking like it does. Should I first be taking the natural log of both sides? If so, what are the logs of sin and cos?

Thanks!

 

[fresh_42]

Do you know what $(e^{iΘ})^2$ is?

 

[SamRoss]

Do you know what $(e^{iΘ})^2$ is?

Oh duh...
(e^iΘ)²=(cosΘ+isinΘ)²=cos²Θ+2isinΘcosΘ-sin²Θ which automatically leads to the answer. I was so stuck on looking at it in exponential form I didn't see what was right in front of me. Thanks for the help!