The image contains a mathematical problem with multiple-choice answers. To solve it effectively, I'll extract and analyze the problem from the image.

Problem Extracted:

The expression is given as:

$\text{Expression} = \frac{1}{(1+i)^n} + \frac{1}{(1-i)^n}$

The problem asks for the simplified form of this expression. The options provided are in terms of trigonometric or exponential functions. Let's solve this.

Solution:

Given the expression:

$\frac{1}{(1+i)^n} + \frac{1}{(1-i)^n}$

Let us first rewrite $1+i$ and $1-i$ in polar form:

$1+i = \sqrt{2} \left(\cos\left(\frac{\pi}{4}\right) + i \sin\left(\frac{\pi}{4}\right)\right)$ $1-i = \sqrt{2} \left(\cos\left(\frac{\pi}{4}\right) - i \sin\left(\frac{\pi}{4}\right)\right)$

Now, raising these to the power $n$:

$(1+i)^n = (\sqrt{2})^n \left(\cos\left(\frac{n\pi}{4}\right) + i \sin\left(\frac{n\pi}{4}\right)\right)$ $(1-i)^n = (\sqrt{2})^n \left(\cos\left(\frac{n\pi}{4}\right) - i \sin\left(\frac{n\pi}{4}\right)\right)$

The original expression becomes:

$\frac{1}{(\sqrt{2})^n \left(\cos\left(\frac{n\pi}{4}\right) + i \sin\left(\frac{n\pi}{4}\right)\right)} + \frac{1}{(\sqrt{2})^n \left(\cos\left(\frac{n\pi}{4}\right) - i \sin\left(\frac{n\pi}{4}\right)\right)}$

Simplifying by multiplying numerator and denominator by the complex conjugate:

$= \frac{\cos\left(\frac{n\pi}{4}\right) - i \sin\left(\frac{n\pi}{4}\right)}{(\sqrt{2})^n \left(\cos^2\left(\frac{n\pi}{4}\right) + \sin^2\left(\frac{n\pi}{4}\right)\right)} + \frac{\cos\left(\frac{n\pi}{4}\right) + i \sin\left(\frac{n\pi}{4}\right)}{(\sqrt{2})^n \left(\cos^2\left(\frac{n\pi}{4}\right) + \sin^2\left(\frac{n\pi}{4}\right)\right)}$

Given that $\cos^2(x) + \sin^2(x) = 1$, the denominator simplifies to $(\sqrt{2})^n$.

The imaginary terms cancel out, so we are left with:

$= \frac{2 \cos\left(\frac{n\pi}{4}\right)}{(\sqrt{2})^n}$

Simplifying:

$= 2 \times \frac{\cos\left(\frac{n\pi}{4}\right)}{2^{n/2}}$

$= 2^{1-\frac{n}{2}} \cos\left(\frac{n\pi}{4}\right)$

Answer:

The simplified expression is:

$2^{1 - \frac{n}{2}} \cos\left(\frac{n\pi}{4}\right)$

This corresponds to option 5 in the image.

Would you like more details or have any questions?
Here are some related questions you might find useful:

1. What are the properties of Euler's formula in complex numbers?
2. How do you convert complex numbers to polar form?
3. What is the significance of roots of unity in complex analysis?
4. Can you explain the geometric interpretation of multiplying complex numbers?
5. How does De Moivre's Theorem simplify the computation of powers of complex numbers?

Tip: When dealing with powers of complex numbers, converting them to polar form often simplifies the calculations considerably.