If `A=[{:(cos q,sin q),(-sin q, cos q):}]` , then variefy that `A^(2)=[{:( cos 2q, sin 2q),(-sin 2q,cos 2 q):}]`

Explanation

We have, `A=[{:(cos q,sin q),(-sin q,cos q):}]`
`therefore A^(2)=A.A=[{:(cos q,sin q),(-sin q,cos q):}].[{:(cos q,sin q),(-sin q,cos q):}]`
`[{:(cos^(2)q-sin^(2)q,cos q sin q+sin q cos q),(-sin q cos q-cos q sin q,-sin^(2)q+cos^(2)q):}]`
`=[{:(cos2q,2sinq cosq),(-2sinq, cos 2q):}] [because cos^(2) theta-sin^(2) theta=cos2 theta]`
`=[{:(cos2q,sin 2q),(-sin2q,cos2q):}][because sin2 theta=2 sin theta-cos theta]`