This example derives the half-angle identities using algebra and the double angles identities.

Start with the double angle identities.

Transform the equations by substituting the half-angle of (alpha) in for (theta).

Simplify the left side of the equations.

Add to both sides of the first equation in step 3. Then subtract from both sides.

Divide both sides by two.

Take the square root of both sides.

This gives us the first half-angle identity.

Rearrange the second equation from step 3 so that the half angle is on the left side.

Add to both sides.

Divide both sides by two.

Take the square root of both sides.

This gives us the second half-angle identity.

In conclusion, the two half angle idenities are given below.