solve the problem

Given : $A=\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]$

$A^{T}=\left[\begin{array}{cc}\cos x & -\sin x \\ \sin x & \cos x\end{array}\right]$

Now,

$A+A^{T}=I$

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

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

$\Rightarrow\left[\begin{array}{cc}2 \cos x & 0 \\ 0 & 2 \cos x\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$

$\Rightarrow 2 \cos x=1$

$\Rightarrow \cos x=\frac{1}{2}$

$\Rightarrow x=\frac{\pi}{3}$