Defining the quantities $\tan\theta = t= s/c$ and $$\cot 2\theta=\tau = \frac{a_{ll}-a_{kk}}{2a_{kl}},$$ we obtain the quadratic equation (using $\cot 2\theta=1/2(\cot \theta-\tan\theta)$ $$t^2+2\tau t-1= 0,$$ resulting in $$t = -\tau \pm \sqrt{1+\tau^2},$$ and $c$ and $s$ are easily obtained via $$c = \frac{1}{\sqrt{1+t^2}},$$ and $s=tc$. Convince yourself that we have $|\theta| \le \pi/4$. This has the effect of minimizing the difference between the matrices $\mathbf{B}$ and $\mathbf{A}$ since $$\mathrm{norm}(\mathbf{B}-\mathbf{A})_F^2=4(1-c)\sum_{i=1,i\ne k,l}^n(a_{ik}^2+a_{il}^2) +\frac{2a_{kl}^2}{c^2}.$$