To determine which equation represents [tex]\(\cot \theta\)[/tex] given [tex]\(\csc \theta = \frac{8}{7}\)[/tex], we will follow these steps.
1. Calculate [tex]\(\sin \theta\)[/tex]:
[tex]\[
\sin \theta = \frac{1}{\csc \theta} = \frac{7}{8}
\][/tex]
2. Calculate [tex]\(\cos \theta\)[/tex] using the Pythagorean identity:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Substituting [tex]\(\sin \theta\)[/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]
Taking the positive square root (since we're dealing with standard trigonometric functions and assume [tex]\(\theta\)[/tex] is in the first quadrant):
[tex]\[
\cos \theta = \sqrt{\frac{15}{64}} = \frac{\sqrt{15}}{8}
\][/tex]
3. Calculate [tex]\(\cot \theta\)[/tex]:
[tex]\[
\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\frac{\sqrt{15}}{8}}{\frac{7}{8}} = \frac{\sqrt{15}}{8} \cdot \frac{8}{7} = \frac{\sqrt{15}}{7}
\][/tex]
Therefore, the correct equation that represents [tex]\(\cot \theta\)[/tex] is:
[tex]\[
\boxed{\cot \theta = \frac{\sqrt{15}}{7}}
\][/tex]
