If `xcostheta=ycos(theta+(2pi)/3)=z cos(theta+(4pi)/3)` , prove that `x y+y z+z x=0.`

Explanation

To prove that \( xy + yz + zx = 0 \) given the equations \( x \cos \theta = y \cos \left( \theta + \frac{2\pi}{3} \right) = z \cos \left( \theta + \frac{4\pi}{3} \right) \), we can follow these steps:

### Step 1: Set a common variable
Let \( k = x \cos \theta = y \cos \left( \theta + \frac{2\pi}{3} \right) = z \cos \left( \theta + \frac{4\pi}{3} \right) \).

### Step 2: Express \( x, y, z \) in terms of \( k \)
From the above, we can express \( x, y, z \) as:
\[
...