What is the value of tan y if `(siny+cosy)/(siny-cosy)=3`?

Options

A. 1
B. `(1)/(sqrt(3))`
C. `(1)/(2)`
D. 2

Answer

To find the value of \( \tan y \) given the equation

\[ \frac{\sin y + \cos y}{\sin y - \cos y} = 3, \]

we can follow these steps:

### Step 1: Rewrite the Equation
Start with the given equation:

\[ \frac{\sin y + \cos y}{\sin y - \cos y} = 3. \]

### Step 2: Cross-Multiply
Cross-multiply to eliminate the fraction:

\[ \sin y + \cos y = 3(\sin y - \cos y). \]

### Step 3: Expand the Right Side
Distributing the 3 on the right side gives:

\[ \sin y + \cos y = 3\sin y - 3\cos y. \]

### Step 4: Rearrange the Equation
Now, rearranging the equation to group the terms involving \( \sin y \) and \( \cos y \):

\[ \sin y + \cos y - 3\sin y + 3\cos y = 0. \]

This simplifies to:

\[ -2\sin y + 4\cos y = 0. \]

### Step 5: Solve for \( \tan y \)
Rearranging gives:

\[ 2\sin y = 4\cos y. \]

Dividing both sides by \( 2\cos y \) (assuming \( \cos y \neq 0 \)) gives:

\[ \tan y = \frac{4}{2} = 2. \]

### Final Answer
Thus, the value of \( \tan y \) is:

\[ \boxed{2}. \]

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