How do you prove #cos2x=cos^2x-sin^2# using other trigonometric identities?
Apply the angle-sum identity for cosine to
The identity needed is the angle-sum identity for cosine.
With that, we have
Alternatively, you can use De Moivre's Theorem of complex numbers to prove the identity.
This maybe is not a very nice proof for the identities themselves for a trigonometry student, but I find it a very useful way to derive the formula if you can't remember it.
De Moivre's Theorem says that if you have a complex number
Exponent of that complex number can be expressed as:
If we let
We can than use De Moivre's theorem to say:
We can also express
These two expressions both equal
Expanding the right hand side, we get:
Since the imaginary parts on the left must equal the imaginary parts on the right and the same for the real, we can deduce the following relationships:
And with that, we've proved both the double angle identities for
In fact, using complex number results to derive trigonometric identities is a quite powerful technique. You can for example prove the angle sum and difference formulas with just a few lines using Euler's identity.