If {cos ^{ - 1}}x + {cos ^{ - 1}}y + {cos ^{ | vedclass

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(C) Given that ${\cos ^{ - 1}}x + {\cos ^{ - 1}}y + {\cos ^{ - 1}}z = 3\pi $. We know that the range of the principal value branch of ${\cos ^{ - 1}}x

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$3$

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(C) Given that ${\cos ^{ - 1}}x + {\cos ^{ - 1}}y + {\cos ^{ - 1}}z = 3\pi $. We know that the range of the principal value branch of ${\cos ^{ - 1}}x$ is $[0, \pi ]$. Therefore,$0 \le {\cos ^{ - 1}}x \le \pi $,$0 \le {\cos ^{ - 1}}y \le \pi $,and $0 \le {\cos ^{ - 1}}z \le \pi $. The sum of three values,each at most $\pi $,can only be $3\pi $ if each individual value is equal to $\pi $. Thus,${\cos ^{ - 1}}x = \pi $,${\cos ^{ - 1}}y = \pi $,and ${\cos ^{ - 1}}z = \pi $. This implies $x = \cos \pi = -1$,$y = \cos \pi = -1$,and $z = \cos \pi = -1$. Substituting these values into the expression $xy + yz + zx$: $xy + yz + zx = (-1)(-1) + (-1)(-1) + (-1)(-1) = 1 + 1 + 1 = 3$.

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

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