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. Recognize that \( \cos^2(-\theta) = \cos^2(\theta) \) because cosine is an even function, meaning it is symmetric about the y-axis.
2. Rewrite \( \csc^2(\theta) \) as \( \frac{1}{\sin^2(\theta)} \) since cosecant is the reciprocal of sine.
3. Substitute \( \tan^2(\theta) = \frac{\sin^2(\theta)}{\cos^2(\theta)} \) into the expression \( \left(\frac{\tan^2\theta}{\sin^2\theta}-1\right) \) to simplify it.
4. Simplify the expression \( \left(\frac{\sin^2(\theta)}{\cos^2(\theta)\sin^2(\theta)} - 1\right) \) to \( \left(\frac{1}{\cos^2(\theta)} - 1\right) \).
5. Combine the simplified terms and use trigonometric identities to further simplify the expression to reach the final result.