Evaluate: `lim_(thetato0)(cos^2(1-cos^2(1-cos^2.....(1-cos^2(theta)))))/(sin(pi(sqrt(theta+4)-2)/theta))`

Options

A. (a) `2`
B. (b) `sqrt(2)`
C. (c) `1/2`
D. (d) `1/(sqrt(2))`

Answer

To evaluate the limit

\[ \lim_{\theta \to 0} \frac{\cos^2(1 - \cos^2(1 - \cos^2(\ldots(1 - \cos^2(\theta))\ldots)))}{\sin\left(\frac{\pi(\sqrt{\theta + 4} - 2)}{\theta}\right)}, \]

we will break it down into steps.

### Step 1: Analyze the Numerator

The numerator is

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

Let's denote

\[ x_0 = \theta, \quad x_{n+1} = 1 - \cos^2(x_n). \]

As \(\theta \to 0\), \(x_0 \to 0\). We need to find the limit of \(x_n\) as \(n \to \infty\).

Using the identity \(1 - \cos^2(x) = \sin^2(x)\), we have:

\[ x_1 = \sin^2(x_0) \approx x_0^2 \quad \text{(for small } x_0\text{)}. \]

Continuing this process, we find:

\[ x_2 = 1 - \cos^2(x_1) \approx 1 - \cos^2(x_0^2) \approx \sin^2(x_0^2) \approx (x_0^2)^2 = x_0^4. \]

This suggests that \(x_n\) approaches \(0\) very quickly. In fact, we can show that \(x_n \approx \theta^{2^n}\) for large \(n\).

Thus, as \(n \to \infty\), \(x_n \to 0\) implies:

\[ \cos^2(x_n) \to \cos^2(0) = 1. \]

### Step 2: Analyze the Denominator

The denominator is

\[ \sin\left(\frac{\pi(\sqrt{\theta + 4} - 2)}{\theta}\right). \]

As \(\theta \to 0\), \(\sqrt{\theta + 4} \to 2\), thus:

\[ \sqrt{\theta + 4} - 2 \to 0. \]

Using the binomial approximation:

\[ \sqrt{\theta + 4} \approx 2 + \frac{\theta}{4} \quad \text{(for small } \theta\text{)}, \]

we find:

\[ \sqrt{\theta + 4} - 2 \approx \frac{\theta}{4}. \]

Substituting this back into the denominator gives:

\[ \frac{\pi(\sqrt{\theta + 4} - 2)}{\theta} \approx \frac{\pi \cdot \frac{\theta}{4}}{\theta} = \frac{\pi}{4}. \]

Thus, we have:

\[ \sin\left(\frac{\pi(\sqrt{\theta + 4} - 2)}{\theta}\right) \to \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}. \]

### Step 3: Combine Results

Now we can combine our results:

- The limit of the numerator is \(1\).
- The limit of the denominator is \(\frac{1}{\sqrt{2}}\).

Thus, the overall limit is:

\[ \lim_{\theta \to 0} \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2}. \]

### Final Answer

The value of the limit is

\[ \sqrt{2}. \]

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