The system is the following:
$\sqrt{\sin w \sin x}+\sqrt{\cos w\cos x}=1\tag1$
$\sqrt{\sin v\cos x}+\sqrt{\sin x\cos v}=4^{1/3}\big(\sin u\sin x\cos v\cos x\big)^{1/12}\tag2$
$\big(\sin u\sin x\big)^{1/4}-\big(\cos u\cos x\big)^{1/4}=\big(\sin v\sin w\big)^{1/4}-\big(\cos v\cos w\big)^{1/4}\tag3$
$\big(\sin u\sin v\sin w\sin x\big)^{1/4}+\big(\cos u\cos v\cos w\cos x\big)^{1/4}+\big(\sin 2u\sin 2v\sin 2w\sin 2x\big)^{1/12}=1\tag4$
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....?