We are asked to find the modulus of the expression:

\[
\frac{2}{1 + \cos\theta + i\sin\theta}
\]

### Step 1: Simplify the denominator
We know from Euler's formula that:

\[
\cos\theta + i\sin\theta = e^{i\theta}
\]

Thus, the denominator becomes:

\[
1 + \cos\theta + i\sin\theta = 1 + e^{i\theta}
\]

### Step 2: Find the modulus
To find the modulus of the fraction, we first find the modulus of the denominator. The modulus of \(1 + e^{i\theta}\) is calculated as:

\[
|1 + e^{i\theta}| = \sqrt{(1 + \cos\theta)^2 + (\sin\theta)^2}
\]

Expanding:

\[
|1 + e^{i\theta}| = \sqrt{(1 + \cos\theta)^2 + \sin^2\theta}
\]

Now, simplify the expression inside the square root:

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

Using the identity \(\cos^2\theta + \sin^2\theta = 1\), we get:

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

Thus, the modulus of the denominator is:

\[
|1 + e^{i\theta}| = \sqrt{2 + 2\cos\theta} = \sqrt{2(1 + \cos\theta)} = \sqrt{4\cos^2\left(\frac{\theta}{2}\right)} = 2|\cos\left(\frac{\theta}{2}\right)|
\]

### Step 3: Find the modulus of the fraction
The modulus of the numerator is simply \(2\), and the modulus of the denominator is \(2|\cos\left(\frac{\theta}{2}\right)|\).

Thus, the modulus of the entire fraction is:

\[
\frac{2}{2|\cos\left(\frac{\theta}{2}\right)|} = \frac{1}{|\cos\left(\frac{\theta}{2}\right)|}
\]

### Final Answer:
The modulus of \(\frac{2}{1 + \cos\theta + i\sin\theta}\) is:

\[
\boxed{\frac{1}{|\cos\left(\frac{\theta}{2}\right)|}}
\]

Do you want details or have any questions?

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Here are 5 related questions to explore:

1. What is the modulus of \( e^{i\theta} \)?
2. How do you find the modulus of complex numbers in general?
3. What is Euler's formula and how is it used in trigonometry?
4. Can you apply this approach to other trigonometric identities?
5. How can you express \(1 + \cos\theta\) in terms of half-angle identities?

**Tip:** The modulus of a complex number \( a + bi \) is \( \sqrt{a^2 + b^2} \).

Discover how to calculate the modulus of the expression 2 / (1 + cos(θ) + i*sin(θ)) using Euler's formula. This problem explores key concepts from complex numbers and trigonometry, suitable for high school students in Grades 10-12.

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