Let A = [ cases cosα & - sinα sinα & cosα cases ], (a ∈ R) such...

∵ A = \left[\begin{cases}cos\alpha & - sin\alpha \\ sin\alpha & cos\alpha\end{cases}\right]

{⇒ A}^{2} = \left[\begin{cases}\cos 2\alpha & - \sin 2\alpha \\ \sin 2\alpha & \cos 2\alpha\end{cases}\right]

⇒ {A}^{3} = \left[\begin{cases}\cos 3\alpha & - \sin 3\alpha \\ \sin 3\alpha & \cos 3\alpha\end{cases}\right]

Similarly, {A}^{32} = \left[\begin{cases}\cos 32\alpha & - \sin 32\alpha \\ \sin 32\alpha & \cos 32\alpha\end{cases}\right]
Given,

{A}^{32} = \left[\begin{cases}0 & - 1 \\ 1 & 0\end{cases}\right]

⇒ \left[\begin{cases}\cos 32\alpha & - \sin 32\alpha \\ \sin 32\alpha & \cos 32\alpha\end{cases}\right] = \left[\begin{cases}0 & - 1 \\ 1 & 0\end{cases}\right]

On comparing $\cos 32\alpha = 0$ & $\sin 32\alpha = 1$, we get,
$\alpha = \dfrac{\pi}{64}$