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!