Question

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

• A. -1
• B. 0
• C. 1
• D. 3

Hint:

Trigonometry Tip: Boundary condition problems are common in entrance exams. If a sum of bounded functions (like $\sin^{-1}, \cos^{-1}, x^2, |x|$) equals their maximum or minimum possible theoretical limits, each individual function MUST be at its respective limit.

Answer 1

The Correct Option is A

Concept: Inverse Trigonometric Functions - Principal Value Branches and Boundary Conditions.

Step 1: Identify the range of the inverse cosine function. The principal value branch (range) of the inverse cosine function, $\cos^{-1}(\theta)$, is defined strictly in the closed interval $[0, \pi]$. This means the maximum possible value for any single $\cos^{-1}$ term is exactly $\pi$.

Step 2: Analyze the given equation based on maximum limits. The given equation is $\cos^{-1}x + \cos^{-1}y + \cos^{-1}z = 3\pi$. Since there are three terms and the maximum value of each term is $\pi$, their sum can only reach $3\pi$ if and only if each individual term is operating at its absolute maximum value.

Step 3: Determine the individual values of the variables. Because of the boundary condition constraint, we can confidently set up three separate equations: $\cos^{-1}x = \pi$, $\cos^{-1}y = \pi$, and $\cos^{-1}z = \pi$.

Step 4: Solve for $x, y,$ and $z$. Take the cosine of both sides for each equation: $x = \cos(\pi)$, $y = \cos(\pi)$, and $z = \cos(\pi)$. Since $\cos(\pi) = -1$, we deduce that $x = y = z = -1$.

Step 5: Evaluate the final algebraic expression. Substitute $x = -1$ into the requested expression: $x^{2025} + x^{2026} + x^{2027}$. This becomes $(-1)^{2025} + (-1)^{2026} + (-1)^{2027}$. A negative base raised to an odd power results in $-1$, and raised to an even power results in $1$. Therefore, the evaluation is $-1 + 1 - 1 = -1$. $$ \therefore \text{The value of the expression is } -1. $$