To find which function is equivalent to the inverse of [tex]\( y = \sec(x) \)[/tex], let's start by understanding the definition of the secant function and how to find its inverse.
1. Definition of Secant Function:
[tex]\[ y = \sec(x) \][/tex]
By definition:
[tex]\[ \sec(x) = \frac{1}{\cos(x)} \][/tex]
Therefore, we have:
[tex]\[ y = \frac{1}{\cos(x)} \][/tex]
2. Rewriting the Secant Function:
To find the inverse, let's express this equation in a way that separates [tex]\( x \)[/tex]:
[tex]\[ y = \frac{1}{\cos(x)} \][/tex]
Which can be rewritten as:
[tex]\[ \cos(x) = \frac{1}{y} \][/tex]
3. Finding the Inverse:
Now, we need to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. Take the arccosine (inverse cosine) of both sides of the equation:
[tex]\[ x = \cos^{-1} \left( \frac{1}{y} \right) \][/tex]
4. Inverse Function:
Therefore, the function equivalent to the inverse of [tex]\( y = \sec(x) \)[/tex] is:
[tex]\[ y = \cos^{-1} \left( \frac{1}{x} \right) \][/tex]
Here, [tex]\( x \)[/tex] serves as the input to the inverse function, so we switch back to the variable [tex]\( y \)[/tex] in the final expression.
Given the functions listed:
[tex]\[
\begin{array}{l}
y=\cot ^{-1}(x) \\
y=\cos ^{-1}\left(\frac{1}{x}\right) \\
y=\csc ^{-1}(x) \\
y=\sin ^{-1}\left(\frac{1}{x}\right)
\end{array}
\][/tex]
The correct choice is:
[tex]\[ y = \cos^{-1} \left( \frac{1}{x} \right) \][/tex]
Thus, the function equivalent to the inverse of [tex]\( y = \sec(x) \)[/tex] is:
[tex]\[ y = \cos^{-1} \left( \frac{1}{x} \right) \][/tex]
Therefore, the answer is the second choice:
[tex]\[ y = \cos^{-1} \left( \frac{1}{x} \right). \][/tex]
