A = cosX sinX
then prove that
An = cos nX sin nX
-sin nX cos nX , n belongs to all natural no.
Let the result be true for n = m. Then,
Now we will show that the result is true for n = m + 1 i.e.,
By the definition of integral powers of a square matrix, we have
This shows that the result is true for n = m + 1, whenever it is true for n = m.
Hence, by the principle of mathematical induction the result is valid for any positive integer n.