To find the value of [tex]\(\left(-e^{\cos^{-1} x}\right)^2\)[/tex], follow these steps:
1. Understand the Inverse Cosine Function: The inverse cosine function, denoted [tex]\(\cos^{-1} x\)[/tex] or [tex]\(\arccos x\)[/tex], gives us an angle [tex]\(\theta\)[/tex] such that [tex]\(\cos\theta = x\)[/tex]. The function [tex]\(\arccos x\)[/tex] maps the interval [tex]\([-1, 1]\)[/tex] to [tex]\([0, \pi]\)[/tex].
2. Exponentiation with Base [tex]\(e\)[/tex]: Given the expression [tex]\(e^{\cos^{-1} x}\)[/tex], we are taking the exponential of the angle [tex]\(\theta\)[/tex].
3. Square of Negative Exponential: The expression we need to evaluate is [tex]\(\left(-e^{\cos^{-1} x}\right)^2\)[/tex]. Squaring a negative number results in a positive number. Thus:
[tex]\[
\left(-e^{\cos^{-1} x}\right)^2 = \left(e^{\cos^{-1} x}\right)^2
\][/tex]
4. Simplify the Expression: Therefore, we can simplify the expression as:
[tex]\[
\left(e^{\cos^{-1} x}\right)^2
\][/tex]
5. Final Result: Recognizing that squaring exponential terms involves squaring the exponent:
[tex]\[
\left(e^{\cos^{-1} x}\right)^2 = e^{2 \cdot \cos^{-1} x}
\][/tex]
Hence, the value of [tex]\(\left(-e^{\cos^{-1} x}\right)^2\)[/tex] simplifies to:
[tex]\[
e^{2 \cdot \cos^{-1} x}
\][/tex]
