# How do you prove #cos2x=cos^2x-sin^2# using other trigonometric identities?

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Apply the angle-sum identity for cosine to

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The identity needed is the angle-sum identity for cosine.

With that, we have

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Alternatively, you can use De Moivre's Theorem of complex numbers to prove the identity.

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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.

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