If `z = (cos theta, sin theta)`, find `(z - 1/z)`

Answer

To solve the problem of finding \( z - \frac{1}{z} \) where \( z = \cos \theta + i \sin \theta \), we will follow these steps:

### Step 1: Write down the expression for \( z \)
Given:
\[ z = \cos \theta + i \sin \theta \]

### Step 2: Find \( \frac{1}{z} \)
To find \( \frac{1}{z} \), we multiply the numerator and denominator by the conjugate of \( z \):
\[ \frac{1}{z} = \frac{1}{\cos \theta + i \sin \theta} \cdot \frac{\cos \theta - i \sin \theta}{\cos \theta - i \sin \theta} \]
This gives:
\[ \frac{1}{z} = \frac{\cos \theta - i \sin \theta}{\cos^2 \theta + \sin^2 \theta} \]
Using the identity \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[ \frac{1}{z} = \cos \theta - i \sin \theta \]

### Step 3: Substitute \( z \) and \( \frac{1}{z} \) into the expression \( z - \frac{1}{z} \)
Now we substitute \( z \) and \( \frac{1}{z} \) into the expression:
\[ z - \frac{1}{z} = \left( \cos \theta + i \sin \theta \right) - \left( \cos \theta - i \sin \theta \right) \]

### Step 4: Simplify the expression
Simplifying the above expression:
\[ z - \frac{1}{z} = \cos \theta + i \sin \theta - \cos \theta + i \sin \theta \]
The \( \cos \theta \) terms cancel out:
\[ z - \frac{1}{z} = 2i \sin \theta \]

### Final Result
Thus, the final result is:
\[ z - \frac{1}{z} = 2i \sin \theta \]

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