Question: If $$\big(\cos x \cos y\big)^{2/3}+\big(\sin x \sin y \big)^{2/3}=1\tag1$$ $$\big(\cos y \cos z\big)^{2/3}+\big(\sin y \sin z \big)^{2/3}=1\tag2$$ then $$\sqrt{3}\big(\sin y \cos y \big)^{1/3}\bigg(\big(\frac {\cos^2{z}}{\cos y}\big)^{2/3}-\big(\frac {\sin^2{z}}{\sin y}\big)^{2/3}\bigg)= \bigg(\big(\frac {\cos^2{y}}{\cos x}\big)^{2/3}-\big(\frac {\sin^2{y}}{\sin x}\big)^{2/3}\bigg) \big((\sin x \cos z)^{2/3}-(\cos x \sin z)^{2/3}\big)\tag3$$ and verify that
$(\sin x)^2=\frac 1 2-\frac {377\sqrt 5}{2662}+\frac{63 \sqrt 15}{1331}$
$(\sin y)^2=\frac {1}{5\sqrt 5\phi^5}$
$(\sin z)^2=\frac 1 2-\frac {377\sqrt 5}{2662}-\frac{63 \sqrt 15}{1331}$
Many other solutions exist.
$\phi$ is the golden ratio.