The Double and Triple Angle Formulas Derivation by de Moivre’s Theorem

And Half Angle Formulas as a Bonus at The End

In the following, the formulas for the tangent will be omitted. This is because they can be easily obtained by using the property
tanθ = sinθ/cosθ.

Let’s review the formulas for double and triple angle.

And here is de Moivre’s theorem that we will use to derive these formulas.

Double Angle Formulas:

Substitute n = 2 into de Moivre’s theorem.

Expand the left side and divide it into real and imaginary parts.

Resubstitute the expanded expression into de Moivre’s theorem for n = 2.