Write each expression in terms of sine and​ cosine, and then simplify so that no quotients appear in the final expression and all functions are of theta only.
StartFraction cosecant squared left parenthesis negative theta right parenthesis minus 1 Over 1 minus cosine squared left parenthesis negative theta right parenthesis EndFraction
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Part 1
StartFraction cosecant squared left parenthesis negative theta right parenthesis minus 1 Over 1 minus cosine squared left parenthesis negative theta right parenthesis EndFraction
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We are tasked with rewriting the expression in terms of sine and cosine, and simplifying it to eliminate quotients and ensure all functions are in terms of \(\theta\).

The given expression is:

\[
\frac{\csc^2(-\theta) - 1}{1 - \cos^2(-\theta)}
\]

### Step 1: Use known identities

1. **Cosecant Identity**: 
   \[
   \csc(-\theta) = \frac{1}{\sin(-\theta)} = \frac{1}{\sin(\theta)}
   \]
   since \(\sin(-\theta) = -\sin(\theta)\), but squaring the function eliminates the negative:
   \[
   \csc^2(-\theta) = \frac{1}{\sin^2(\theta)}
   \]

2. **Cosine Identity**:
   \[
   \cos(-\theta) = \cos(\theta)
   \]
   Therefore, \(\cos^2(-\theta) = \cos^2(\theta)\).

Using these, the expression becomes:

\[
\frac{\csc^2(\theta) - 1}{1 - \cos^2(\theta)}
\]

### Step 2: Apply further identities

- Recall that \(\csc^2(\theta) - 1 = \cot^2(\theta)\):
  \[
  \frac{\cot^2(\theta)}{1 - \cos^2(\theta)}
  \]

- Additionally, we know that \(1 - \cos^2(\theta) = \sin^2(\theta)\) (this is the Pythagorean identity).

Thus, the expression becomes:

\[
\frac{\cot^2(\theta)}{\sin^2(\theta)}
\]

### Step 3: Express in terms of sine and cosine

Recall that \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\), so \(\cot^2(\theta) = \frac{\cos^2(\theta)}{\sin^2(\theta)}\). Substituting this into the expression:

\[
\frac{\frac{\cos^2(\theta)}{\sin^2(\theta)}}{\sin^2(\theta)} = \frac{\cos^2(\theta)}{\sin^4(\theta)}
\]

### Final Answer:

The simplified expression is:

\[
\frac{\cos^2(\theta)}{\sin^4(\theta)}
\]

This is the expression entirely in terms of sine and cosine, with no quotients involving trigonometric functions left.

Would you like further details on this or have any questions?

---

### Related questions:
1. What is the identity for \(\csc^2(\theta)\)?
2. How do the even and odd properties of trigonometric functions affect the identities?
3. Can you show how to simplify \(\frac{1 - \sin^2(\theta)}{\cos^2(\theta)}\)?
4. What is the Pythagorean identity involving sine and cosine?
5. How do we prove \(\csc^2(\theta) - 1 = \cot^2(\theta)\)?

### Tip:
Always remember the fundamental trigonometric identities (like the Pythagorean identities) when simplifying expressions involving sine and cosine. They can save a lot of time!

Write each expression in terms of sine and​ cosine, and then simplify so that no quotients appear in the final expression and all functions are of theta only. StartFraction cosecant squared left parenthesis negative theta right parenthesis minus 1 Over 1 minus cosine squared left parenthesis negative theta right parenthesis EndFraction

Learn how to simplify the trigonometric expression involving cosecant and cosine functions in terms of sine and cosine. This step-by-step solution breaks down the use of trigonometric identities, such as the Pythagorean identity, making it easier for students in Grades 10-12 to understand.

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