For a complex number `z` the minimum value of `|z|+|z-cos alpha-i sin alpha|` (where `i=sqrt-1`) is:

Options

A. 0
B. 1
C. 2
D. None of these

Answer

To find the minimum value of the expression \( |z| + |z - (\cos \alpha + i \sin \alpha)| \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ |z| + |z - (\cos \alpha + i \sin \alpha)| \]
This can be interpreted as the sum of the distance from the point represented by the complex number \( z \) to the origin (0) and the distance from \( z \) to the point \( (\cos \alpha, \sin \alpha) \) on the unit circle.

### Step 2: Use the triangle inequality
According to the triangle inequality, we know that:
\[ |z| + |z - (\cos \alpha + i \sin \alpha)| \geq |(\cos \alpha + i \sin \alpha)| \]
This means that the sum of the distances is at least the distance from the origin to the point \( (\cos \alpha, \sin \alpha) \).

### Step 3: Calculate the distance
The distance from the origin to the point \( (\cos \alpha, \sin \alpha) \) is given by:
\[ |(\cos \alpha + i \sin \alpha)| = \sqrt{\cos^2 \alpha + \sin^2 \alpha} \]
Using the Pythagorean identity, we know that:
\[ \cos^2 \alpha + \sin^2 \alpha = 1 \]
Thus, we have:
\[ |(\cos \alpha + i \sin \alpha)| = \sqrt{1} = 1 \]

### Step 4: Establish the minimum value
From the triangle inequality, we established that:
\[ |z| + |z - (\cos \alpha + i \sin \alpha)| \geq 1 \]
The minimum value of \( |z| + |z - (\cos \alpha + i \sin \alpha)| \) occurs when \( z \) is exactly at the point \( (\cos \alpha, \sin \alpha) \). In this case, both distances are equal to 1, leading to:
\[ |z| + |z - (\cos \alpha + i \sin \alpha)| = 1 + 0 = 1 \]

### Conclusion
Thus, the minimum value of \( |z| + |z - (\cos \alpha + i \sin \alpha)| \) is:
\[ \boxed{1} \]