If `z=i log(2-sqrt(3))` then cosz

Options

A. i
B. 2i
C. 1
D. 2

Answer

To solve the problem \( z = i \log(2 - \sqrt{3}) \) and find \( \cos z \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the expression for \( \cos z \)**:
The formula for \( \cos z \) in terms of exponential functions is:
\[ \cos z = \frac{e^{iz} + e^{-iz}}{2} \]

2. **Substitute \( z \)**:
Substitute \( z = i \log(2 - \sqrt{3}) \) into the formula:
\[ \cos z = \frac{e^{i(i \log(2 - \sqrt{3}))} + e^{-i(i \log(2 - \sqrt{3}))}}{2} \]
This simplifies to:
\[ \cos z = \frac{e^{-\log(2 - \sqrt{3})} + e^{\log(2 - \sqrt{3})}}{2} \]

3. **Simplify the exponentials**:
Recall that \( e^{\log a} = a \) for any positive \( a \):
\[ e^{-\log(2 - \sqrt{3})} = \frac{1}{2 - \sqrt{3}} \quad \text{and} \quad e^{\log(2 - \sqrt{3})} = 2 - \sqrt{3} \]
Therefore, we can rewrite \( \cos z \):
\[ \cos z = \frac{\frac{1}{2 - \sqrt{3}} + (2 - \sqrt{3})}{2} \]

4. **Combine the terms**:
To combine the fractions, we need a common denominator:
\[ \cos z = \frac{1 + (2 - \sqrt{3})(2 - \sqrt{3})}{2(2 - \sqrt{3})} \]

5. **Expand and simplify**:
First, calculate \( (2 - \sqrt{3})^2 \):
\[ (2 - \sqrt{3})^2 = 4 - 4\sqrt{3} + 3 = 7 - 4\sqrt{3} \]
Now substitute this back:
\[ \cos z = \frac{1 + (7 - 4\sqrt{3})}{2(2 - \sqrt{3})} = \frac{8 - 4\sqrt{3}}{2(2 - \sqrt{3})} \]

6. **Simplify further**:
Simplifying the numerator:
\[ \cos z = \frac{4(2 - \sqrt{3})}{2(2 - \sqrt{3})} = 2 \]

### Final Result:
Thus, the value of \( \cos z \) is:
\[ \cos z = 2 \]