$x, y , z$ are respectively the $sines$ and $p, q, r$ are respectively the $cosines$ of the .....

Question

$x, y , z$ 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.

Answer 1

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 $\cos\left(2A-\dfrac{4\pi}3\right)+\cos\left(2A+\dfrac{4\pi}3\right)=2\cos2A\cos\dfrac{4\pi}3=?$