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The problem asks to express \(\csc(t)\) in terms of \(\cot(t)\) given that the terminal point of \(t\) is in Quadrant III. Let's work through this step by step.

### Given:
- The expression to solve is \(\csc(t)\) in terms of \(\cot(t)\).
- The angle \(t\) lies in **Quadrant III**.
- We know that \(\csc(t) = \frac{1}{\sin(t)}\) and \(\cot(t) = \frac{\cos(t)}{\sin(t)}\).

### Key Trigonometric Relationships:
In general:
\[
\sin^2(t) + \cos^2(t) = 1
\]
In terms of \(\cot(t)\):
\[
\cot^2(t) = \frac{\cos^2(t)}{\sin^2(t)} \Rightarrow \sin^2(t) = \frac{1}{1 + \cot^2(t)}
\]

### Step-by-Step Solution:
1. **Find \(\sin(t)\) in terms of \(\cot(t)\):**
   \[
   \sin^2(t) = \frac{1}{1 + \cot^2(t)}
   \]
   \[
   \sin(t) = \pm \frac{1}{\sqrt{1 + \cot^2(t)}}
   \]
   
   Since we are in **Quadrant III**, where sine is negative, we choose the negative root:
   \[
   \sin(t) = -\frac{1}{\sqrt{1 + \cot^2(t)}}
   \]

2. **Express \(\csc(t)\):**
   By definition, \(\csc(t) = \frac{1}{\sin(t)}\). Substituting the expression for \(\sin(t)\):
   \[
   \csc(t) = -\sqrt{1 + \cot^2(t)}
   \]

Thus, the answer is:
\[
\csc(t) = -\sqrt{1 + \cot^2(t)}
\]

### Would you like more details or have any questions about this solution?

Here are some related questions for further exploration:
1. How does the sign of \(\csc(t)\) change in each quadrant?
2. What is the relationship between \(\sec(t)\) and \(\tan(t)\)?
3. Can you derive \(\cos(t)\) in terms of \(\cot(t)\) in Quadrant III?
4. How does the Pythagorean identity apply to other trigonometric functions?
5. What is the significance of the sign of \(\cot(t)\) in different quadrants?

**Tip:** Always consider the quadrant when determining the sign of trigonometric functions!

Write the first expression in terms of the second if the terminal point determined by t is in the given quadrant: csc(t), cot(t); Quadrant III.

Learn how to express csc(t) in terms of cot(t) when the terminal point lies in Quadrant III. This solution covers key trigonometric relationships and the Pythagorean identity with a focus on determining the correct sign for functions in different quadrants. Suitable for students in Grades 10-12.

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