De Moivre's Theorem states that:
(cosx+isinx)^n=cosnx+isinnx
This can be used (preferably with binomial expansion for higher values) to find double, triple (etc) angle rules by substituting in different values of n and then equating the real and imaginary parts of both sides.
Example:
Let n=3
Therefore, (cosx+isinx)^3=cos3x+isin3x
cos^3x+3cos^2x(isinx)+3cosx(isinx)^2+(isinx)^3=cos3x+isin3x
cos^3x+3icos^2xsinx-3cosxsin^2x-isin^3x=cos3x+isin3x
You can now equate the Re and Im parts of both sides, the idea here being that if you have two equal complex numbers:
a+bi=c+di => a=c and b=d since the two domains can't interfere. It's a similar idea to vectors in more than one dimension, how different components in different dimensions can't interfere.
Therefore, Equating Re:
cos^3x-3cosxsin^2x=cos3x
Equating Im:
3cos^2xsinx-sin^3x=sin3x
These rules, especially for n=2, make computing integrals like intsin^2xdx significantly easier.
