Certainly! Let's solve this step by step.
Given [tex]\(\csc \theta = \frac{8}{7}\)[/tex], we need to find [tex]\(\cot \theta\)[/tex].
1. Express [tex]\(\sin \theta\)[/tex] in terms of [tex]\(\csc \theta\)[/tex]:
[tex]\[
\sin \theta = \frac{1}{\csc \theta} = \frac{1}{\frac{8}{7}} = \frac{7}{8}
\][/tex]
2. Use the Pythagorean identity to find [tex]\(\cos \theta\)[/tex]:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
[tex]\[
\left( \frac{7}{8} \right)^2 + \cos^2 \theta = 1
\][/tex]
[tex]\[
\frac{49}{64} + \cos^2 \theta = 1
\][/tex]
[tex]\[
\cos^2 \theta = 1 - \frac{49}{64}
\][/tex]
[tex]\[
\cos^2 \theta = \frac{64}{64} - \frac{49}{64}
\][/tex]
[tex]\[
\cos^2 \theta = \frac{15}{64}
\][/tex]
[tex]\[
\cos \theta = \sqrt{\frac{15}{64}} = \frac{\sqrt{15}}{8}
\][/tex]
3. Calculate [tex]\(\cot \theta\)[/tex] using [tex]\(\cos \theta\)[/tex] and [tex]\(\sin \theta\)[/tex]:
[tex]\[
\cot \theta = \frac{\cos \theta}{\sin \theta}
\][/tex]
[tex]\[
\cot \theta = \frac{\frac{\sqrt{15}}{8}}{\frac{7}{8}}
\][/tex]
Simplify the fraction:
[tex]\[
\cot \theta = \frac{\sqrt{15}}{8} \times \frac{8}{7} = \frac{\sqrt{15}}{7}
\][/tex]
Thus, the equation that represents [tex]\(\cot \theta\)[/tex] is:
[tex]\[
\cot \theta = \frac{\sqrt{15}}{7}
\][/tex]
Therefore, the correct choice is:
[tex]\[
\boxed{\frac{\sqrt{15}}{7}}
\][/tex]
