Question

If $\cos^{-1} x + \cos^{-1} y + \cos^{-1} z = 3\pi$, then $x(y + z) + y(z + x) + z(x + y)$ equals to:

• A. 0
• B. 1
• C. 6
• D. 12

Answer 1

The Correct Option is C

Given $\cos^{-1} x + \cos^{-1} y + \cos^{-1} z = 3\pi$, it follows that $x = y = z = -1$ because $\cos^{-1}(-1) = \pi$.

Substituting $x = y = z = -1$, we have: \[ x(y + z) + y(z + x) + z(x + y) = (-1)(-1 - 1) + (-1)(-1 - 1) + (-1)(-1 - 1) = 6. \]