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 rewriting the given expression: \( \left( \frac{\tan^2\theta}{\sin^2\theta} - 1 \right) \csc^2(\theta) \cos^2(-\theta) \).
2. Recall the trigonometric identities: \( \tan^2\theta = \frac{\sin^2\theta}{\cos^2\theta} \), \( \csc^2\theta = \frac{1}{\sin^2\theta} \), and \( \cos^2(-\theta) = \cos^2\theta \).
3. Substitute these identities into the expression: \( \left( \frac{\frac{\sin^2\theta}{\cos^2\theta}}{\sin^2\theta} - 1 \right) \frac{1}{\sin^2\theta} \cos^2\theta \).
4. Simplify the expression inside the parentheses: \( \frac{1}{\cos^2\theta} - 1 \).
5. Recognize that \( \frac{1}{\cos^2\theta} - 1 = \tan^2\theta \), and simplify the entire expression to find the result.