Suppose you want cos(nx) and sin(nx) in terms of sin(x). You'll need cos(x) as well which can be found using
First note that
Raising both sides to the nth power:
And note that
Therefore, by the transitive property
So
and
Expanding using the binomial theorem yields:
Splitting it into real and imaginary parts:
And
First note that
Raising both sides to the nth power:
And note that
Therefore, by the transitive property
So
and
Expanding using the binomial theorem yields:
Splitting it into real and imaginary parts:
And
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