The expression given is $\sin^2(\theta)$, and you want to find its derivative with respect to $\theta$.

To differentiate $\sin^2(\theta)$, we apply the chain rule. Here's the step-by-step process:

1. Rewrite $\sin^2(\theta)$ as $(\sin(\theta))^2$.
2. Use the chain rule: $\frac{d}{d\theta} \left( (\sin(\theta))^2 \right) = 2\sin(\theta) \cdot \frac{d}{d\theta} (\sin(\theta))$.
3. The derivative of $\sin(\theta)$ with respect to $\theta$ is $\cos(\theta)$.

Now, putting it all together: $\frac{d}{d\theta} \left( \sin^2(\theta) \right) = 2\sin(\theta) \cdot \cos(\theta)$

Thus, the derivative of $\sin^2(\theta)$ is: $2\sin(\theta)\cos(\theta)$

Alternatively, you could express this result using the double-angle identity for sine: $2\sin(\theta)\cos(\theta) = \sin(2\theta)$

So, the derivative of $\sin^2(\theta)$ can also be written as: $\sin(2\theta)$

Would you like more details or explanations? Here are 5 related questions you might find interesting:

1. What is the derivative of $\cos^2(\theta)$?
2. How do you apply the chain rule to more complex functions?
3. What is the derivative of $\tan^2(\theta)$?
4. How do double-angle identities help in simplifying trigonometric derivatives?
5. What is the integral of $\sin^2(\theta)$?

Tip: Always check if trigonometric identities can simplify your expressions before or after differentiating.