If A = [ cases cosθ & isinθ isinθ & cosθ cases ], (θ = (π)/(24))...

{A}^{2} = \left[$$\begin{array}{ll}cos\theta isin\theta \\ isin\theta cos\theta\end{array}$$\right]\left[$$\begin{array}{ll}cos\theta isin\theta \\ isin\theta cos\theta\end{array}$$\right]$</p><p>$= \left[$$\begin{array}{ll}\cos^{2}\theta - \sin^{2}\theta2isin\theta cos\theta \\ 2isin\theta cos\theta\cos^{2}\theta - \sin^{2}\theta\end{array}$$\right]$</p><p>$= \left[$$\begin{array}{ll}\cos 2\theta isin2\theta \\ isin2\theta cos2\theta\end{array}$$\right]$</p><p>${A}^{3} = \left[$$\begin{array}{ll}\cos 2\theta isin2\theta \\ isin2\theta cos2\theta\end{array}$$\right]\left[$$\begin{array}{ll}cos\theta isin\theta \\ isin\theta cos\theta\end{array}$$\right]$</p><p>$= \left[$$\begin{array}{ll}\cos 3\theta isin3\theta \\ isin3\theta cos3\theta\end{array}$$\right]$</p><p>Similarly,${A}^{5} = \left[$$\begin{array}{ll}\cos 5\theta isin5\theta \\ isin5\theta cos5\theta\end{array}$$\right] = \left[$$\begin{array}{ll}ab \\ cd\end{array}$$\right]$</p><p>$a = \cos 5\theta$,$b = isin5\theta$,$c = isin5\theta$</p><p>$d = \cos 5\theta$</p><p>$a = d, b = c$</p><p>$( 1 ) {a}^{2} + {b}^{2} = \cos^{2}5\theta - \sin^{2}5\theta = \cos 10\theta \in $\left(0, 1\right)$ as $\theta = \dfrac{\pi}{24}$

$( 2 )$ ${a}^{2} - {d}^{2} = 0$

$( 3 )$ ${a}^{2} - {b}^{2} = \cos^{2}5\theta + \sin^{2}5\theta = 1$

$( 4 )$ ${a}^{2} - {c}^{2} = {a}^{2} - {b}^{2} = 1$