Find  
 
 
 
\[\displaystyle\lim_{\theta\to\scriptsize\dfrac{\pi}{2}}\dfrac{\sin^2(2\theta)}{1-\sin^2(\theta)}\].
Choose 1 answer:
Choose 1 answer:

We are tasked with finding the limit

\[
\lim_{\theta \to \frac{\pi}{2}} \frac{\sin^2(2\theta)}{1 - \sin^2(\theta)}.
\]

### Step 1: Simplify the expression

First, note that \( \sin^2(2\theta) \) can be rewritten using the double-angle identity for sine:

\[
\sin(2\theta) = 2\sin(\theta)\cos(\theta) \quad \Rightarrow \quad \sin^2(2\theta) = 4\sin^2(\theta)\cos^2(\theta).
\]

Thus, the original limit becomes:

\[
\lim_{\theta \to \frac{\pi}{2}} \frac{4\sin^2(\theta)\cos^2(\theta)}{1 - \sin^2(\theta)}.
\]

### Step 2: Analyze the denominator

Using the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \), we can rewrite the denominator as:

\[
1 - \sin^2(\theta) = \cos^2(\theta).
\]

So the limit simplifies further to:

\[
\lim_{\theta \to \frac{\pi}{2}} \frac{4\sin^2(\theta)\cos^2(\theta)}{\cos^2(\theta)}.
\]

### Step 3: Simplify the expression

Cancel \( \cos^2(\theta) \) from the numerator and denominator:

\[
\lim_{\theta \to \frac{\pi}{2}} 4\sin^2(\theta).
\]

### Step 4: Evaluate the limit

As \( \theta \to \frac{\pi}{2} \), we know that:

\[
\sin\left(\frac{\pi}{2}\right) = 1.
\]

Thus, the limit becomes:

\[
4 \times 1^2 = 4.
\]

### Final Answer:
\[
\boxed{4}.
\]

Would you like more details on any of the steps?

Here are 5 related questions to consider:
1. What is the double-angle identity for cosine and how could it be used in limits?
2. How do Pythagorean identities simplify trigonometric expressions?
3. What are common limit evaluation strategies for trigonometric functions near special angles like \(\frac{\pi}{2}\)?
4. How do limits behave when expressions involve trigonometric powers?
5. How would the limit change if the angle approached 0 instead of \(\frac{\pi}{2}\)?

**Tip:** When evaluating trigonometric limits, it helps to recall standard identities and limits like \( \lim_{\theta \to 0} \frac{\sin(\theta)}{\theta} = 1 \).

\lim_{\theta\to\frac{\pi}{2}}\frac{\sin^2(2\theta)}{1-\sin^2(\theta)}

Learn how to solve the trigonometric limit \lim_{\theta\to\frac{\pi}{2}} \frac{\sin^2(2\theta)}{1-\sin^2(\theta)} step by step. This problem explores key trigonometric identities such as the double-angle identity for sine and Pythagorean identities, making it suitable for college-level calculus students.

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