<s>[INST] $x, y , z$ are respectively the $sines$ and $p, q, r$ are respectively the $cosines$ of the ..... $x, y , z$ are respectively the $sines$ and $p, q, r$ are respectively the $cosines$ of the angles $\alpha, \beta, \gamma$, which are in A.P. with common difference $\frac{2\pi}{3}$. 
1. $yz + zx + xy = ?$ 
2. $x^2 (qy - rz) + y^2 (r - px) + z^2 (px - qy) = ?$
For the first I tried using the formula for $2sinAsinB$ but it is not helping. How do I solve them? Please, help.
 [/INST] WLOG  let $x=\sin\left(A-\dfrac{2\pi}3\right),y=\sin A,z=\sin\left(A+\dfrac{2\pi}3\right)$
$$2(xy+yz+zx)=(x+y+z)^2-(x^2+y^2+z^2)$$
We can prove $x+y+z=0$
Using $\cos2B=1-2\sin^2B,$
$$2(x^2+y^2+z^2)=3-\left\{\cos\left(2A-\dfrac{4\pi}3\right)+\cos2A+\cos\left(2A+\dfrac{4\pi}3\right)\right\}$$
Now $\cos\left(2A-\dfrac{4\pi}3\right)+\cos\left(2A+\dfrac{4\pi}3\right)=2\cos2A\cos\dfrac{4\pi}3=?$ 
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