(5 - 2 sin ^ 2 theta cos ^ 2 theta) / sin ^ 2 theta cos ^ 2 theta

We are given the following expression:

\[
\frac{5 - 2 \sin^2 \theta \cos^2 \theta}{\sin^2 \theta \cos^2 \theta}
\]

Let's break it down step by step:

1. **Separate the terms in the numerator**:
   The numerator has two terms, \(5\) and \(-2 \sin^2 \theta \cos^2 \theta\). We can split the fraction accordingly:
   \[
   \frac{5}{\sin^2 \theta \cos^2 \theta} - \frac{2 \sin^2 \theta \cos^2 \theta}{\sin^2 \theta \cos^2 \theta}
   \]

2. **Simplify the second term**:
   The second term simplifies directly because the denominator is the same as the numerator:
   \[
   \frac{2 \sin^2 \theta \cos^2 \theta}{\sin^2 \theta \cos^2 \theta} = 2
   \]

   So the expression becomes:
   \[
   \frac{5}{\sin^2 \theta \cos^2 \theta} - 2
   \]

3. **Use a trigonometric identity**:
   Recall that:
   \[
   \sin^2 \theta \cos^2 \theta = \frac{1}{4} \sin^2 2\theta
   \]
   Using this identity, we rewrite the first term:
   \[
   \frac{5}{\sin^2 \theta \cos^2 \theta} = \frac{5}{\frac{1}{4} \sin^2 2\theta} = \frac{20}{\sin^2 2\theta}
   \]

4. **Final simplified form**:
   The expression simplifies to:
   \[
   \frac{20}{\sin^2 2\theta} - 2
   \]

Thus, the simplified form of the given expression is:

\[
\frac{20}{\sin^2 2\theta} - 2
\]

Would you like more details on any of these steps or have any questions?

Here are 5 related questions you might find interesting:
1. How do trigonometric identities simplify complex expressions?
2. What is the significance of double-angle identities in trigonometry?
3. How can trigonometric functions be used to solve real-world problems?
4. What are some common mistakes when working with trigonometric simplifications?
5. How does the Pythagorean identity help in simplifying trigonometric expressions?

**Tip**: Memorizing key trigonometric identities, such as double-angle and Pythagorean identities, can significantly speed up solving problems involving trigonometric functions.

(5 - 2 sin^2 theta cos^2 theta) / sin^2 theta cos^2 theta

This guide shows how to simplify the expression (5 - 2 sin^2 theta cos^2 theta) / sin^2 theta cos^2 theta using trigonometric identities such as the double-angle formula. The step-by-step solution includes key simplifications and the use of identities suitable for Grades 10-12 students.

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