If cos ^{-1} x+cos ^{-1} y+cos ^{-1} z=3 pi,t | vedclass

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(A) Given that $\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=3 \pi$. Since the range of $\cos ^{-1} 	heta$ is $[0, \pi]$,the maximum value of each term is

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(A) Given that $\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=3 \pi$. Since the range of $\cos ^{-1} 	heta$ is $[0, \pi]$,the maximum value of each term is $\pi$. For the sum of three terms to be $3 \pi$,each term must be equal to its maximum value: $\cos ^{-1} x = \pi$,$\cos ^{-1} y = \pi$,and $\cos ^{-1} z = \pi$. This implies $x = \cos \pi = -1$,$y = \cos \pi = -1$,and $z = \cos \pi = -1$. Now,substitute $x = -1$ into the expression $x^{2025}+x^{2026}+x^{2027}$: $(-1)^{2025} + (-1)^{2026} + (-1)^{2027} = -1 + 1 - 1 = -1$.

If $\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=3 \pi$,then the value of $x^{2025}+x^{2026}+x^{2027}$ is

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