To determine which expression is equivalent to [tex]\(\cot \theta\)[/tex] given that [tex]\(\tan \theta = -\frac{3}{8}\)[/tex], let's break down the problem step-by-step.
1. Understanding Definitions:
- [tex]\(\tan \theta\)[/tex] is the ratio of the opposite side to the adjacent side in a right triangle.
- [tex]\(\cot \theta\)[/tex] is the reciprocal of [tex]\(\tan \theta\)[/tex], meaning [tex]\(\cot \theta = \frac{1}{\tan \theta}\)[/tex].
2. Given:
[tex]\[
\tan \theta = -\frac{3}{8}
\][/tex]
3. Finding [tex]\(\cot \theta\)[/tex]:
[tex]\[
\cot \theta = \frac{1}{\tan \theta} = \frac{1}{-\frac{3}{8}}
\][/tex]
4. Simplifying the Expression:
[tex]\[
\frac{1}{-\frac{3}{8}} = -\frac{8}{3}
\][/tex]
Thus, [tex]\(\cot \theta\)[/tex] is [tex]\(-\frac{8}{3}\)[/tex].
5. Matching with the Given Options:
- Option 1: [tex]\(\frac{1}{-\frac{3}{8}}\)[/tex] simplifies directly to [tex]\(-\frac{8}{3}\)[/tex], which is equal to [tex]\(\cot \theta\)[/tex].
- Option 2: [tex]\(-\frac{3}{8} + 1\)[/tex] results in a different value and does not equal [tex]\(\cot \theta\)[/tex].
- Option 3: [tex]\(\sqrt{1 + \left(-\frac{8}{3}\right)^2}\)[/tex] is a different form and does not simplify to [tex]\(\cot \theta\)[/tex].
- Option 4: [tex]\(\left(-\frac{3}{8}\right)^2 + 1\)[/tex] again results in a different value and does not equal [tex]\(\cot \theta\)[/tex].
Therefore, the expression equivalent to [tex]\(\cot \theta\)[/tex] is:
[tex]\[
\frac{1}{-\frac{3}{8}}
\][/tex]
So the correct answer is:
[tex]\[
\boxed{\frac{1}{-\frac{3}{8}}}
\][/tex]
