https://file.aiquickdraw.com/mathbot/file/1729338261438pnegwi25.jpg

The expression in the image is as follows:

\[
(1 + i)^n + (1 - i)^n = 2 \cdot \frac{n+1}{2} \cdot \cos\left(\frac{n\pi}{4}\right)
\]

This is a combination of complex numbers and trigonometric functions. To analyze it:

### Key Components:
1. \( (1 + i)^n \) and \( (1 - i)^n \): These represent powers of complex numbers. We can simplify them using **De Moivre's theorem**, which expresses powers of complex numbers in terms of trigonometric functions.
   
2. \( \cos\left(\frac{n\pi}{4}\right) \): This is a cosine function evaluated at angles that are multiples of \( \frac{\pi}{4} \), which corresponds to common angles on the unit circle like \( 45^\circ \), \( 90^\circ \), etc.

### Simplification Strategy:

- Using De Moivre's theorem:
  \[
  (1+i) = \sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right)
  \]
  So:
  \[
  (1 + i)^n = \left(\sqrt{2}\right)^n \cdot \left(\cos\left(\frac{n\pi}{4}\right) + i\sin\left(\frac{n\pi}{4}\right)\right)
  \]
  Similarly, for \( (1 - i)^n \):
  \[
  (1 - i)^n = \left(\sqrt{2}\right)^n \cdot \left(\cos\left(\frac{n\pi}{4}\right) - i\sin\left(\frac{n\pi}{4}\right)\right)
  \]

- Adding \( (1+i)^n \) and \( (1-i)^n \), the imaginary parts cancel, and we are left with:
  \[
  (1 + i)^n + (1 - i)^n = 2 \cdot \left(\sqrt{2}\right)^n \cdot \cos\left(\frac{n\pi}{4}\right)
  \]

From the equation in the image:
\[
2 \cdot \left(\sqrt{2}\right)^n \cdot \cos\left(\frac{n\pi}{4}\right) = 2 \cdot \frac{n+1}{2} \cdot \cos\left(\frac{n\pi}{4}\right)
\]

Canceling out the cosine terms and simplifying:
\[
\left(\sqrt{2}\right)^n = \frac{n+1}{2}
\]

Thus, this is the resulting equation that relates the power of \( \sqrt{2} \) with \( n \).

Would you like further details on solving this equation or have any other questions?

### 5 Related Questions:
1. How does De Moivre's theorem work for complex numbers?
2. What are the geometric interpretations of \( (1+i)^n \) on the complex plane?
3. Can this equation help find specific values of \( n \)?
4. How can we graph the cosine function for multiples of \( \frac{\pi}{4} \)?
5. How is the equation \( \left(\sqrt{2}\right)^n = \frac{n+1}{2} \) solved for \( n \)?

### Tip:
When dealing with powers of complex numbers, using polar forms and De Moivre's theorem often simplifies calculations dramatically.

(1 + i)^n + (1 - i)^n = 2 * (n+1)/2 * cos(nπ/4)

Explore how to solve the equation (1 + i)^n + (1 - i)^n = 2 * (n+1)/2 * cos(nπ/4) using complex numbers, De Moivre's theorem, and trigonometry. Suitable for undergraduate students studying complex analysis.

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