Question

If $\cos ^ { - 1 } x + \cos ^ { - 1 } y + \cos ^ { - 1 } z = 3 \pi$ then $x y + y z + z x =$

• A. 0
• B. 1
• C. 3
• D. –3

Answer 1

Answer: (C)

3

Answer 2

Given $\cos ^ { - 1 } x + \cos ^ { - 1 } y + \cos ^ { - 1 } z = 3 \pi$

$\because 0 \leq \cos ^ { - 1 } x \leq \pi$

$\therefore 0 \leq \cos ^ { - 1 } y \leq \pi$and $0 \leq \cos ^ { - 1 } z \leq \pi$

Here $\cos ^ { - 1 } x = \cos ^ { - 1 } y = \cos ^ { - 1 } z = \pi$

$\Rightarrow x = y = z = \cos \pi = - 1$

∴ $x y + y z + z x$ $= ( - 1 ) ( - 1 ) + ( - 1 ) ( - 1 ) + ( - 1 ) ( - 1 )$

$= 1 + 1 + 1 = 3$.