If `(sin(x+y))/(sin(x-y))=(a+b)/(a-b)`, then what is `(tanx)/(tany)` equal to?

Options

A. `b/a`
B. `a/b`
C. ab
D. 1

Answer

To solve the equation \(\frac{\sin(x+y)}{\sin(x-y)} = \frac{a+b}{a-b}\), we need to manipulate the equation step by step to find the value of \(\frac{\tan x}{\tan y}\).

### Step 1: Apply the Sum-to-Product Identities
We start with the left-hand side of the equation:
\[ \frac{\sin(x+y)}{\sin(x-y)} \]
Using the sum-to-product identities, we can express \(\sin(x+y)\) and \(\sin(x-y)\) as:
\[ \sin(x+y) = \sin x \cos y + \cos x \sin y \]
\[ \sin(x-y) = \sin x \cos y - \cos x \sin y \]
Thus, we rewrite the left-hand side:
\[ \frac{\sin x \cos y + \cos x \sin y}{\sin x \cos y - \cos x \sin y} \]

### Step 2: Simplify the Expression
Now, we can simplify the expression:
\[ \frac{\sin x \cos y + \cos x \sin y}{\sin x \cos y - \cos x \sin y} = \frac{\sin x \cos y + \cos x \sin y}{\sin x \cos y - \cos x \sin y} \]
Let’s denote \(\sin x \cos y\) as \(A\) and \(\cos x \sin y\) as \(B\). Therefore, we have:
\[ \frac{A + B}{A - B} \]

### Step 3: Set Equal to Right-Hand Side
Now equate this to the right-hand side:
\[ \frac{A + B}{A - B} = \frac{a+b}{a-b} \]

### Step 4: Cross Multiply
Cross-multiplying gives us:
\[ (A + B)(a - b) = (A - B)(a + b) \]
Expanding both sides:
\[ Aa - Ab + Ba - Bb = Aa + Ab - Ba - Bb \]

### Step 5: Rearranging Terms
Rearranging the terms leads to:
\[ 2Ba = 2Ab \]
Thus, we can simplify this to:
\[ Ba = Ab \]
Assuming \(B \neq 0\) and \(A \neq 0\), we can divide both sides by \(AB\):
\[ \frac{B}{A} = \frac{a}{b} \]

### Step 6: Relate to Tangents
Recall that:
\[ \frac{B}{A} = \frac{\cos x \sin y}{\sin x \cos y} = \frac{\tan y}{\tan x} \]
Thus, we can write:
\[ \frac{\tan x}{\tan y} = \frac{b}{a} \]

### Final Result
Therefore, we find that:
\[ \frac{\tan x}{\tan y} = \frac{b}{a} \]