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To prove the identity:

\[
\frac{\sec \theta \sin \theta}{\tan \theta + \cot \theta} = \sin^2 \theta
\]

we'll work with each part of the expression separately. Let's proceed with simplifying the left-hand side.

### Step 1: Rewrite the Expression
The left side of the equation is:

\[
\frac{\sec \theta \sin \theta}{\tan \theta + \cot \theta}
\]

Recall the definitions of secant, tangent, and cotangent:
- \(\sec \theta = \frac{1}{\cos \theta}\)
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
- \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)

### Step 2: Substitute these Definitions
Substitute \(\sec \theta\), \(\tan \theta\), and \(\cot \theta\) into the expression:

\[
\frac{\frac{1}{\cos \theta} \cdot \sin \theta}{\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta}}
\]

Simplify the numerator:

\[
\frac{\sin \theta}{\cos \theta}
\]

Now, simplify the denominator by finding a common denominator:

\[
\frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta}
\]

### Step 3: Use the Pythagorean Identity
We know that \(\sin^2 \theta + \cos^2 \theta = 1\), so the denominator simplifies to:

\[
\frac{1}{\sin \theta \cos \theta}
\]

Now we have:

\[
\frac{\frac{\sin \theta}{\cos \theta}}{\frac{1}{\sin \theta \cos \theta}}
\]

### Step 4: Simplify the Complex Fraction
Dividing by a fraction is equivalent to multiplying by its reciprocal:

\[
\frac{\sin \theta}{\cos \theta} \cdot \frac{\sin \theta \cos \theta}{1} = \sin^2 \theta
\]

### Conclusion
The left-hand side simplifies to \(\sin^2 \theta\), which matches the right-hand side. Thus, we have proven the identity:

\[
\frac{\sec \theta \sin \theta}{\tan \theta + \cot \theta} = \sin^2 \theta
\]

Would you like further details on any of the steps, or do you have any questions?

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Here are 5 related questions:
1. How can the Pythagorean identity help simplify trigonometric expressions?
2. What is the significance of rewriting trigonometric identities using basic functions like sine and cosine?
3. How do you approach proving a trigonometric identity?
4. How does understanding reciprocal identities help in simplifying complex trigonometric expressions?
5. What are some other commonly used trigonometric identities?

**Tip:** When proving identities, try converting all functions into sines and cosines, as it often makes simplification easier.

Prove the identity: \( \frac{\sec \theta \sin \theta}{\tan \theta + \cot \theta} = \sin^2 \theta \)

Learn how to prove the trigonometric identity \( \frac{\sec \theta \sin \theta}{\tan \theta + \cot \theta} = \sin^2 \theta \) with a step-by-step solution. This solution covers key trigonometric concepts, including reciprocal identities and the Pythagorean identity. Suitable for high school students in Grades 10-12.

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