Let and (the determinant of T). Furthermore, let a+d (the trace of T) equal for some angle theta. Finally, let . Show that for all natural numbers q, where I is the identity matrix.
The usual way to prove such a formula is by using the principle of mathematical induction. Since the formula is trivially true when , all we need to do is prove that is we assume the formula is true for q (the induction hypothesis) , then it is also true for q+1. I.e. we need to prove that . We begin:
So, we will be done if we can show that . But this is equivalent to . Applying the appropriate “product to sum” trigonometric identity implies the result.