Let

Prove that

* Proof. * The proof is by induction. For we have

These equalities follow from the co-relations of sine and cosine (Theorem 2.3 part (d) on page 96 of Apostol). Thus, the formulas are true for the case . Assume then that they are true for some . For we then have

Similarly, for we have

Therefore, the theorem follows by induction for all positive integers