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Rule

# Double-Angle Identities

The double-angle identities are special cases of the angle sum identities. To evaluate trigonometric functions of the double of an angle, the following identities can be applied.

These identities are useful to find the exact value of the sine, cosine, or tangent at a given angle. For example, knowing that the exact value of can be found by using the second formula for the cosine of the double of an angle. In this case, the first step is to rewrite as Below, a proof of the first two identities is shown. The rest of the identities can be proven by following a similar reasoning.

### Proof

Proof

Start by writing the Angle Sum Identity for sine and cosine. Let and With this, becomes Then, the two formulas above can be rewritten in terms of The Commutative Property of Multiplication can be applied to the second term of the right-hand side of the first equation. Then, by adding the terms on the right-hand side of this equation, the formula for is obtained.

Finally, the Product of Powers Property can be used to rewrite the right-hand side of the second equation. By doing this, the first identity for the cosine of the double of an angle is obtained.