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We are given the following information:

\[
w_1 = e^{ix} \quad \text{and} \quad w_2 = e^{i\left(x - \frac{\pi}{2}\right)}
\]

We are tasked with finding \( w_1 + w_2 \) in the form \( e^{ix}(c + id) \).

### Step-by-Step Solution:

1. **Express \( w_1 \) and \( w_2 \) using Euler's formula:**

   Using Euler's formula, \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \), we can express \( w_1 \) and \( w_2 \) as follows:

   \[
   w_1 = e^{ix} = \cos(x) + i\sin(x)
   \]
   \[
   w_2 = e^{i\left(x - \frac{\pi}{2}\right)} = \cos\left(x - \frac{\pi}{2}\right) + i\sin\left(x - \frac{\pi}{2}\right)
   \]

2. **Simplify \( w_2 \):**

   Recall the trigonometric identities:
   \[
   \cos\left(x - \frac{\pi}{2}\right) = \sin(x)
   \]
   \[
   \sin\left(x - \frac{\pi}{2}\right) = -\cos(x)
   \]

   Therefore:
   \[
   w_2 = \sin(x) - i\cos(x)
   \]

3. **Add \( w_1 \) and \( w_2 \):**

   Now we compute \( w_1 + w_2 \):

   \[
   w_1 + w_2 = \left(\cos(x) + i\sin(x)\right) + \left(\sin(x) - i\cos(x)\right)
   \]

   Combine real and imaginary parts:

   \[
   w_1 + w_2 = \left(\cos(x) + \sin(x)\right) + i\left(\sin(x) - \cos(x)\right)
   \]

4. **Factor out \( e^{ix} \):**

   We want to express the result in the form \( e^{ix}(c + id) \). Notice that:

   \[
   w_1 + w_2 = e^{ix} \left( 1 + \frac{w_2}{w_1} \right)
   \]

   This gives: 
    ***
   
 Do you have any questions?



Let w_1 = e^(ix) and w_2 = e^(i(x - π/2)), where x ∈ R. Find w_1 + w_2 in the form e^(ix)(c + id).

Solve the problem of adding w_1 = e^(ix) and w_2 = e^(i(x - π/2)) in the form e^(ix)(c + id) using Euler's formula and trigonometric identities. This exercise explores complex numbers and is suitable for undergraduate or advanced high school students.

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