Consider the two complex numbers z and w, such that
`w=(z-1)/(z+2)=a+ib, " where " a,b in R " and " i=sqrt(-1).`
If `z=CiStheta`, which of the following does hold good?

Options

A. `sin theta=(9b)/(1-4a)`
B. `cos theta=(1-5a)/(1+4a)`
C. `(1+5a)^(2)+(3b)^(2)=(1-4a)^(2)`
D. All of these

Answer

To solve the problem, we need to analyze the given complex numbers and their relationships. We start with the expression for \( w \) in terms of \( z \).

### Step 1: Rewrite \( z \) in terms of \( \theta \)
Given that \( z = \text{CiS} \theta \), we can express this in rectangular form:
\[ z = \cos \theta + i \sin \theta \]

### Step 2: Substitute \( z \) into the equation for \( w \)
We have:
\[ w = \frac{z - 1}{z + 2} \]
Substituting \( z \):
\[ w = \frac{(\cos \theta + i \sin \theta) - 1}{(\cos \theta + i \sin \theta) + 2} \]
This simplifies to:
\[ w = \frac{(\cos \theta - 1) + i \sin \theta}{(\cos \theta + 2) + i \sin \theta} \]

### Step 3: Multiply numerator and denominator by the conjugate of the denominator
To simplify \( w \), we multiply the numerator and denominator by the conjugate of the denominator:
\[ w = \frac{((\cos \theta - 1) + i \sin \theta)((\cos \theta + 2) - i \sin \theta)}{((\cos \theta + 2)^2 + \sin^2 \theta)} \]

### Step 4: Expand the numerator
Expanding the numerator:
\[ = \frac{(\cos \theta - 1)(\cos \theta + 2) + \sin^2 \theta + i(\sin \theta(\cos \theta + 2) - \sin \theta(\cos \theta - 1))}{(\cos \theta + 2)^2 + \sin^2 \theta} \]
This simplifies to:
\[ = \frac{(\cos^2 \theta + 2\cos \theta - \cos \theta - 2) + \sin^2 \theta + i(3\sin \theta)}{(\cos \theta + 2)^2 + \sin^2 \theta} \]
Using the identity \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[ = \frac{1 + \cos \theta - 2 + i(3\sin \theta)}{(\cos \theta + 2)^2 + \sin^2 \theta} \]
Thus:
\[ = \frac{-1 + \cos \theta + i(3\sin \theta)}{(\cos \theta + 2)^2 + \sin^2 \theta} \]

### Step 5: Identify real and imaginary parts
Let \( w = a + ib \), where:
\[ a = \frac{-1 + \cos \theta}{(\cos \theta + 2)^2 + \sin^2 \theta} \]
\[ b = \frac{3\sin \theta}{(\cos \theta + 2)^2 + \sin^2 \theta} \]

### Step 6: Analyze the relationships
From the expressions for \( a \) and \( b \), we can derive relationships between \( \cos \theta \) and \( \sin \theta \). We can also analyze the conditions under which these hold true.

### Conclusion
The relationships derived from \( a \) and \( b \) can be used to analyze the conditions that hold true for \( z \) and \( w \).