Simplify the expression.
$\left(\frac{\tan^2\theta}{\sin^2\theta}-1\right)\csc^2\left(\theta\right)\cos^2\left(-\theta\right)$

• A. $\cot^2\theta$
• B. $\tan\theta$
• C. 1
• D. – 1

Answer: (C)

Verified step by step guidance

1. Start by recognizing that \( \cos^2(-\theta) = \cos^2(\theta) \) because cosine is an even function.
2. Rewrite \( \csc^2(\theta) \) as \( \frac{1}{\sin^2(\theta)} \).
3. Substitute \( \tan^2(\theta) = \frac{\sin^2(\theta)}{\cos^2(\theta)} \) into the expression.
4. Simplify the expression \( \left( \frac{\tan^2(\theta)}{\sin^2(\theta)} - 1 \right) \csc^2(\theta) \cos^2(\theta) \) by substituting the identities and simplifying.
5. Recognize that the expression simplifies to \( 1 - 1 \) after canceling terms, which results in 0.