Let $x$, $y$, $z$ be the sines of $\alpha$, $\beta$, $\gamma$, and let $p$, $q$, $r$ be the respective cosines, where the angles are in Arithmetic Progression with common difference $\frac{2\pi}{3}$.
Find value of
$$x^2\left(qy-rz\right)+y^2\left(rz-px\right)+z^2\left(px-qy\right)$$
My Attempt: Let $\beta=\left(\alpha+\frac{2\pi}{3}\right)$ and $\gamma=\left(\alpha+\frac{4\pi}{3}\right)$
Given that $$x=\sin\alpha \qquad p=\cos\alpha$$ $$y=\sin\left(\alpha+\frac{2\pi}{3}\right) \qquad q=\cos\left(\alpha+\frac{2\pi}{3}\right)$$ $$z=\sin\left(\alpha+\frac{4\pi}{3}\right) \qquad r=\cos\left(\alpha+\frac{4\pi}{3}\right)$$
Clearly, $$x^2+p^2=y^2+q^2=z^2+r^2=1$$
Also, $$x+y+z=p+q+r=0$$
Now, there must be an algebraic identity which can help solve the problem.