If `x=ycos((2pi)/3)=zcos((4pi)/3)`, then xy+yz+zx=

Options

A. `-1`
B. 0
C. 1
D. 2

Answer

To solve the equation \( x = y \cos\left(\frac{2\pi}{3}\right) = z \cos\left(\frac{4\pi}{3}\right) \) and find the value of \( xy + yz + zx \), we can follow these steps:

### Step 1: Evaluate the cosine values
First, we need to find the values of \( \cos\left(\frac{2\pi}{3}\right) \) and \( \cos\left(\frac{4\pi}{3}\right) \).

- \( \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \)
- \( \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2} \)

### Step 2: Substitute the cosine values into the equations
Now substitute these values into the equations:

\[ x = y \left(-\frac{1}{2}\right) \quad \text{and} \quad x = z \left(-\frac{1}{2}\right) \]

This gives us:

\[ x = -\frac{y}{2} \quad \text{and} \quad x = -\frac{z}{2} \]

### Step 3: Express \( y \) and \( z \) in terms of \( x \)
From the equations above, we can express \( y \) and \( z \) in terms of \( x \):

\[ y = -2x \quad \text{and} \quad z = -2x \]

### Step 4: Substitute \( y \) and \( z \) into \( xy + yz + zx \)
Now we can substitute \( y \) and \( z \) into the expression \( xy + yz + zx \):

\[ xy + yz + zx = x(-2x) + (-2x)(-2x) + (-2x)x \]

### Step 5: Simplify the expression
Let's simplify each term:

1. \( xy = -2x^2 \)
2. \( yz = 4x^2 \)
3. \( zx = -2x^2 \)

Now combine these:

\[ xy + yz + zx = -2x^2 + 4x^2 - 2x^2 = 0 \]

### Final Result
Thus, the value of \( xy + yz + zx \) is:

\[ \boxed{0} \]