If `tantheta=(xsinphi)/(1-xcosphi) and tanphi=(ysintheta)/(1-ycostheta),` then `x/y=` (A) `sinphi/sintheta` (B) `sintheta/sinphi` (C) `sinphi/(1-costheta)` (D) `sintheta/(1-cosphi)`

Answer

To solve the given problem, we start with the two equations provided:

1. \( \tan \theta = \frac{x \sin \phi}{1 - x \cos \phi} \)
2. \( \tan \phi = \frac{y \sin \theta}{1 - y \cos \theta} \)

We need to find the value of \( \frac{x}{y} \).

### Step 1: Rewrite the first equation in terms of cotangent
From the first equation, we can express \( \tan \theta \) as follows:

\[ \tan \theta = \frac{x \sin \phi}{1 - x \cos \phi} \]

Taking the cotangent of both sides, we have:

\[ \cot \theta = \frac{1 - x \cos \phi}{x \sin \phi} \]

### Step 2: Rearranging the cotangent expression
We can rewrite the cotangent expression:

\[ \cot \theta = \frac{1}{x \sin \phi} - \frac{\cos \phi}{\sin \phi} \]

This simplifies to:

\[ \cot \theta + \frac{\cos \phi}{\sin \phi} = \frac{1}{x \sin \phi} \]

### Step 3: Write this as an equation
Now we can express this as:

\[ \cot \theta + \cot \phi = \frac{1}{x \sin \phi} \]

### Step 4: Rewrite the second equation in terms of cotangent
Now, let's work with the second equation:

\[ \tan \phi = \frac{y \sin \theta}{1 - y \cos \theta} \]

Taking the cotangent, we get:

\[ \cot \phi = \frac{1 - y \cos \theta}{y \sin \theta} \]

### Step 5: Rearranging the second cotangent expression
Similar to the first equation, we can rearrange this:

\[ \cot \phi = \frac{1}{y \sin \theta} - \frac{\cos \theta}{\sin \theta} \]

This leads to:

\[ \cot \phi + \cot \theta = \frac{1}{y \sin \theta} \]

### Step 6: Equating the two expressions
From the two equations we derived, we have:

\[ \frac{1}{x \sin \phi} = \cot \theta + \cot \phi \]

and

\[ \frac{1}{y \sin \theta} = \cot \theta + \cot \phi \]

Since both expressions equal \( \cot \theta + \cot \phi \), we can set them equal to each other:

\[ \frac{1}{x \sin \phi} = \frac{1}{y \sin \theta} \]

### Step 7: Cross-multiplying to find \( \frac{x}{y} \)
Cross-multiplying gives us:

\[ y \sin \theta = x \sin \phi \]

Rearranging this, we find:

\[ \frac{x}{y} = \frac{\sin \theta}{\sin \phi} \]

### Final Answer
Thus, the value of \( \frac{x}{y} \) is:

\[ \frac{x}{y} = \frac{\sin \theta}{\sin \phi} \]

The correct option is (B) \( \frac{\sin \theta}{\sin \phi} \).