This example derives the double angle identities using algebra and the sum of two angles identities.
Start with the sum of two angles identities.
Substitute and into the identities. This is same as saying the angle (alpha) is equal to (beta).
Combine the arguments on the left and simplify the expressions on the right. This gives us the double angle identities.
Optionally, the Pythagorean identity, shown below, can be used to calculate the two double-angle identity variations.
Subtract from both sides.
Substitute this expression into the identity from step 3 and combine like terms. This gives us the first variant.
The second variant is found by subtracting from both sides of the pythagorean identity.
Substitute this expression into the identity from step 3 and combine like terms. This gives us the second variant.