Double-Angle Identities<

Double-Angle Identities

For any angle or value \(A\), the following relationships are always true.

\[\sin(2A) = 2\sin(A)\cos(A)\]

\[\begin{align*} \cos(2A) &= 2\cos^2(A) - 1 \\ \cos(2A) &= 1 - 2\sin^2(A) \\ \cos(2A) &= \cos^2(A) - \sin^2(A) \end{align*}\]

\[\tan(2A) = \frac{2\tan(A)}{1-\tan^2(A)}\]

For the double-angle identity of cosine, there are 3 variations of the formula. You can choose whichever is more relevant or more helpful to a specific problem. They are all related through the Pythagorean Theorem.

Simplifying Expressions

As we add in a new set of identities into the mix, remember that the overall process is still the same. Ask yourself, is there an identity I can use or algebra I can do? Let's look at some more examples that use these new identities.

Verify Identities

Just like simplifying, the process to verify an identity hasn't changed even though we have more identities. Let's look at a few examples.

Trigonometric Identities