$A=\begin{bmatrix} cos \alpha & sin \alpha \\ -sin \alpha & cos \alpha \end{bmatrix}$
$A^{t}=A^{'}=\begin{bmatrix} cos \alpha & -sin \alpha \\ sin \alpha & cos \alpha \end{bmatrix}$
$A+A^{'}=\begin{bmatrix} cos \alpha & sin \alpha \\ -sin \alpha & cos \alpha \end{bmatrix}+\begin{bmatrix} cos \alpha &- sin \alpha \\ sin \alpha & cos \alpha \end{bmatrix}$
$A+A^{'}=\begin{bmatrix} 2cos \alpha & 0 \\ 0 & 2cos \alpha \end{bmatrix}$
Put $\alpha =\frac{\pi}{3}$
$A+A^{'}=\begin{bmatrix} 2 \times \frac{1}{2} & 0 \\ 0 & 2 \times \frac{1}{2} \end{bmatrix}$
$A+A^{'}=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}=I$
This is also true for $\frac{5\pi}{3}$ .
The period of the $cos \alpha$ function is $2 \pi$ so values will repeat every $2 \pi$ radians in both directions.
Hence,
$\alpha =\frac{\pi}{3} \pm 2 \pi n, \frac{5\pi }{3} \pm 2 \pi n$