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 \( \lambda = x \cos \theta = y \cos\left(\theta + \frac{2\pi}{3}\right) = z \cos\left(\theta + \frac{4\pi}{3}\right) \).

### Step 2: Express each variable in terms of \( \lambda \)
From the definitions, we can express \( x \), \( y \), and \( z \) as:
\[
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