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How do you derive the double angle identities $\sin{2A}$ and $\cos{2A}$ ?
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If sin(a+b)=sinacosb+ cosasinb.

For sin 2a, we let a=b then we can write the formula as sin(2a)∴sin2a = sin(a+a)=sinacosa+cosasina

Similarly, for cos(a+b)=cosacosb−sinasinb. If we let a=b we have:

cos(2a) =cos(a+a)=cosacosa−sinasina

Therefore cos⁡2a=cos⁡a^2 −sin⁡a^2

Using the square identity, sina^2 + cosa^2=, we can also derive the following formulae:

cos2a=(1−sina^2)−sina^2=1−2sina^2

And finally,

cos2a =cosa^2−(1−cosa^2)=cosa^2−1+cosa^2 = 2cosa^2 − 1
by Diamond (42,490 points)

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