Calculate the derivative of sine squared theta

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.

Learn how to calculate the derivative of sin^2(θ) using the chain rule and trigonometric identities. This step-by-step solution covers important calculus and trigonometric concepts, making it ideal for students studying calculus in high school or college.

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