If A = and A-1 = A’,

$A=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]$ and $A^{\prime}=\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]$

Also,

$A^{-1}=A^{\prime}$

$A A^{-1}=A A^{\prime}$

$I=A A^{\prime}$

$\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]$

$\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=\left[\begin{array}{cc}\cos ^{2} \alpha+\sin ^{2} \alpha & 0 \\ 0 & \sin ^{2} \alpha+\cos ^{2} \alpha\end{array}\right]$

By using equality of matrices, we get

cos2 α + sin2 α = 1, which is true for all real values of α.