Write the expression in terms of sine and​ cosine, and then simplify so that no quotients appear in the final expression.

negative sine squared theta left parenthesis negative 1 minus cotangent squared theta right parenthesis−sin2 θ−1−cot2 θ

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Part 1

Choose the correct answer below.

A.StartFraction cosine squared theta Over sine squared theta EndFraction

StartFraction cosine squared theta Over sine squared theta EndFractioncos2 θsin2 θ

B.negative 1

negative 1−1

C.tangent squared theta

tangent squared thetatan2 θ

D.1

1

E.cotangent squared theta

cotangent squared thetacot2 θ

F.negative StartFraction cosine squared theta Over sine squared theta EndFraction

negative StartFraction cosine squared theta Over sine squared theta EndFraction

To rewrite the expression \(-\sin^2 \theta (-1 - \cot^2 \theta) - \sin^2 \theta - 1 - \cot^2 \theta\) in terms of sine and cosine, let's break it down step by step.

1. Recall that \(\cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta}\).

2. Substitute \(\cot^2 \theta\) into the expression:
   \[
   -\sin^2 \theta \left(-1 - \frac{\cos^2 \theta}{\sin^2 \theta}\right) - \sin^2 \theta - 1 - \frac{\cos^2 \theta}{\sin^2 \theta}
   \]

3. Distribute \(-\sin^2 \theta\):
   \[
   \sin^2 \theta + \frac{\sin^2 \theta \cos^2 \theta}{\sin^2 \theta} - \sin^2 \theta - 1 - \frac{\cos^2 \theta}{\sin^2 \theta}
   \]
   This simplifies to:
   \[
   \sin^2 \theta + \cos^2 \theta - \sin^2 \theta - 1 - \frac{\cos^2 \theta}{\sin^2 \theta}
   \]

4. Since \(\sin^2 \theta + \cos^2 \theta = 1\), we can substitute:
   \[
   1 - \sin^2 \theta - 1 - \frac{\cos^2 \theta}{\sin^2 \theta}
   \]
   Which simplifies to:
   \[
   -\sin^2 \theta - \frac{\cos^2 \theta}{\sin^2 \theta}
   \]

5. Factor out \(-1\):
   \[
   -\left(\sin^2 \theta + \frac{\cos^2 \theta}{\sin^2 \theta}\right)
   \]
   This is:
   \[
   -\left(\frac{\sin^4 \theta + \cos^2 \theta}{\sin^2 \theta}\right)
   \]

6. Since we are asked to remove quotients, let's express it all under the same denominator:
   \[
   -\left(\frac{\sin^4 \theta + \cos^2 \theta}{\sin^2 \theta}\right)
   \]

The final expression is simplified to:

\[
-\frac{\cos^2 \theta}{\sin^2 \theta}
\]

So the answer is:

**F.** \(-\frac{\cos^2 \theta}{\sin^2 \theta}\)

If you have any questions or need further clarification, feel free to ask!

### Related Questions:
1. What are the Pythagorean identities related to sine and cosine?
2. How can you derive \(\cot^2 \theta\) in terms of sine and cosine?
3. What happens to the expression if you change the sign of \(\theta\)?
4. Can you explain how to convert cotangent to sine and cosine?
5. What are some applications of these trigonometric identities?

**Tip:** Remember that simplifying expressions can often involve substituting identities to reduce complexity.

Write the expression in terms of sine and cosine, and then simplify so that no quotients appear in the final expression: -sin^2 θ (-1 - cot^2 θ) - sin^2 θ - 1 - cot^2 θ.

Explore the simplification of the trigonometric expression -sin^2 θ (-1 - cot^2 θ) - sin^2 θ - 1 - cot^2 θ into sine and cosine without quotients. This detailed solution includes key formulas and identities suitable for high school students.

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