`(sin(x+y))/(sin(x-y))=(a+b)/(a-b)`, then `(tanx)/(tany)=?`

Options

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

Answer

To solve the equation \(\frac{\sin(x+y)}{\sin(x-y)} = \frac{a+b}{a-b}\) and find \(\frac{\tan x}{\tan y}\), we can follow these steps:

### Step 1: Apply the Sine Addition and Subtraction Formulas
We start by applying the sine addition and subtraction formulas:
\[ \sin(x+y) = \sin x \cos y + \cos x \sin y \]
\[ \sin(x-y) = \sin x \cos y - \cos x \sin y \]
Thus, we can rewrite the left side of the equation:
\[ \frac{\sin(x+y)}{\sin(x-y)} = \frac{\sin x \cos y + \cos x \sin y}{\sin x \cos y - \cos x \sin y} \]

### Step 2: Rewrite the Equation
Now, we have:
\[ \frac{\sin x \cos y + \cos x \sin y}{\sin x \cos y - \cos x \sin y} = \frac{a+b}{a-b} \]

### Step 3: Cross-Multiply
Cross-multiplying gives us:
\[ (\sin x \cos y + \cos x \sin y)(a-b) = (\sin x \cos y - \cos x \sin y)(a+b) \]

### Step 4: Expand Both Sides
Expanding both sides:
\[ a \sin x \cos y - b \sin x \cos y + a \cos x \sin y - b \cos x \sin y = a \sin x \cos y + b \sin x \cos y - a \cos x \sin y - b \cos x \sin y \]

### Step 5: Rearranging Terms
Rearranging the terms leads to:
\[ a \sin x \cos y - b \sin x \cos y + a \cos x \sin y - b \cos x \sin y - a \sin x \cos y - b \sin x \cos y + a \cos x \sin y + b \cos x \sin y = 0 \]
This simplifies to:
\[ -2b \sin x \cos y + 2a \cos x \sin y = 0 \]

### Step 6: Factor Out Common Terms
Factoring out the common terms gives:
\[ 2(a \cos x \sin y - b \sin x \cos y) = 0 \]
Thus, we have:
\[ a \cos x \sin y = b \sin x \cos y \]

### Step 7: Dividing Both Sides
Dividing both sides by \(\sin y \cos y\) (assuming \(\sin y \neq 0\) and \(\cos y \neq 0\)):
\[ \frac{a}{b} = \frac{\sin x}{\sin y} \cdot \frac{\cos y}{\cos x} \]

### Step 8: Expressing \(\frac{\tan x}{\tan y}\)
We can express \(\frac{\tan x}{\tan y}\) as:
\[ \frac{\tan x}{\tan y} = \frac{\sin x / \cos x}{\sin y / \cos y} = \frac{\sin x \cos y}{\sin y \cos x} \]
From the previous step, we have:
\[ \frac{\tan x}{\tan y} = \frac{a}{b} \]

### Final Result
Thus, the final result is:
\[ \frac{\tan x}{\tan y} = \frac{a}{b} \]

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