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

1. Understand the problem:

Given cos⁻¹x + cos⁻¹y + cos⁻¹z = 3π, we need to find the value of x(y+z) + y(z+x) + z(x+y).

2. Analyze the given equation:

The maximum value of cos⁻¹θ for any θ ∈ [-1,1] is π. The sum equals 3π only if each term equals π.

Thus, cos⁻¹x = cos⁻¹y = cos⁻¹z = π ⇒ x = y = z = -1.

3. Substitute x = y = z = -1:

The expression becomes:

(-1)(-1-1) + (-1)(-1-1) + (-1)(-1-1) = (-1)(-2) + (-1)(-2) + (-1)(-2) = 2 + 2 + 2 = 6

Correct Answer: (C) 6