The system is the following:
$\sqrt{\sin w \sin x}+\sqrt{\cos w\cos x}=1\tag1$
$\sqrt{\sin v\cos x}+\sqrt{\sin x\cos v}=4^{1/3}\big(\sin v\sin x\cos v\cos x\big)^{1/12}\tag2$
$\big(\sin u\sin x\big)^{1/4}-\big(\cos u\cos x\big)^{1/4}=\big(\sin v\sin w\big)^{1/4}-\big(\cos v\cos w\big)^{1/4}\tag3$
$\big(\sin u\sin v\sin w\sin x\big)^{1/4}+\big(\cos u\cos v\cos w\cos x\big)^{1/4}+\big(\sin 2u\sin 2v\sin 2w\sin 2x\big)^{1/12}=1\tag4$
The system of 4 equation above has many solutions, but One of the most beautiful solutions is:
$\sin u=(2-\sqrt{3})(\sqrt{3}-\sqrt{2})^{2}(\frac{\sqrt{5}+\sqrt{3}}{\sqrt{2}})^{2}(\sqrt{6}+\sqrt{5})$ $\sin v=(2+\sqrt{3})(\sqrt{3}-\sqrt{2})^{2}(\frac{\sqrt{5}+\sqrt{3}}{\sqrt{2}})^{2}(\sqrt{6}-\sqrt{5})$ $\sin w=(2+\sqrt{3})(\sqrt{3}-\sqrt{2})^{2}(\frac{\sqrt{5}-\sqrt{3}}{\sqrt{2}})^{2}(\sqrt{6}+\sqrt{5})$ $\sin x=(2-\sqrt{3})(\sqrt{3}-\sqrt{2})^{2}(\frac{\sqrt{5}-\sqrt{3}}{\sqrt{2}})^{2}(\sqrt{6}-\sqrt{5})$
Many others exist! And the best....?
Another solution:
$\sin{u}=(Abc)^{2}d$
$\sin{v}=(aBc)^{2}D$
$\sin{w}=(aBC)^{2}d$
$\sin{x}=(ABC)^{2}D$
$$Aa=Bb=Cc=Dd=1$$
$$a=\frac{\sqrt{3}+1}{\sqrt{2}}$$
$$b=\sqrt{3}+\sqrt{2}$$
$$c=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{2}}$$
$$d=\sqrt{6}+\sqrt{5}$$
Another solution, very simple, is the following:
$\sin x=\cos u=\frac{-\sqrt{30}+3\sqrt{10}-\sqrt{6}-\sqrt{2}}{16}$
$\sin u=\cos x=\frac{+\sqrt{30}+3\sqrt{10}+\sqrt{6}-\sqrt{2}}{16}$
$\sin v=\cos w=\frac{+\sqrt{30}+3\sqrt{10}-\sqrt{6}+\sqrt{2}}{16}$
$\sin w=\cos v=\frac{-\sqrt{30}+3\sqrt{10}+\sqrt{6}+\sqrt{2}}{16}$