A = cosX sinX

-sinX cosX

then prove that

A^{n} = 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*.

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