To solve this problem, let's analyze the expression given:

Expression to simplify:
$\sin^2 \theta (1 + \cot^2 \theta)$

Step-by-Step Solution

1. Rewrite $\cot^2 \theta$ in terms of sine and cosine: $\cot \theta = \frac{\cos \theta}{\sin \theta}$ So, $\cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta}$

2. Substitute $\cot^2 \theta$ into the expression: $\sin^2 \theta \left(1 + \frac{\cos^2 \theta}{\sin^2 \theta}\right)$

3. Simplify the expression inside the parentheses: $1 + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin^2 \theta}$

   Using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$, we get: $1 + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{1}{\sin^2 \theta}$

4. Rewrite the entire expression: $\sin^2 \theta \cdot \frac{1}{\sin^2 \theta}$

5. Simplify by canceling $\sin^2 \theta$: $\sin^2 \theta \cdot \frac{1}{\sin^2 \theta} = 1$

Conclusion

The expression simplifies to $1$.

Answer

The correct answer is:

• C. $1$

Would you like further details on any specific steps?

Here are five questions for practice:

1. What is the Pythagorean identity involving $\sin^2 \theta$ and $\cos^2 \theta$?
2. How can you express $\cot^2 \theta$ in terms of $\sin \theta$ and $\cos \theta$?
3. Why does $\sin^2 \theta \cdot \frac{1}{\sin^2 \theta}$ simplify to $1$?
4. What other trigonometric identities can help simplify complex expressions?
5. How would the solution change if the original expression were $\sin^2 \theta (1 - \cot^2 \theta)$?

Tip:

When simplifying trigonometric expressions, look for identities that can help convert terms into sine and cosine, as this often makes simplification easier.