If tan ` theta = (x)/(y) , " then " (x sin theta + y cos theta)/( x sin theta - y cos theta)` is equal to

Options

A. `(x^(2) + y^(2))/( x^(2) - y^(2))`
B. ` (x^(2) - y^(2))/(x^(2) + y^(2))`
C. `(x)/(sqrt(x^(2) + y^(2)))`
D. ` (y)/( sqrt(x^(2) + y^(2)))`

Answer

To solve the problem where \( \tan \theta = \frac{x}{y} \) and we need to find the value of

\[ \frac{x \sin \theta + y \cos \theta}{x \sin \theta - y \cos \theta}, \]

we can follow these steps:

### Step 1: Express \( \tan \theta \) in terms of sine and cosine
Since \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we can rewrite \( \tan \theta \) as:

\[ \frac{\sin \theta}{\cos \theta} = \frac{x}{y}. \]

### Step 2: Rewrite \( \sin \theta \) and \( \cos \theta \)
From the equation \( \tan \theta = \frac{x}{y} \), we can express \( \sin \theta \) and \( \cos \theta \) in terms of \( x \) and \( y \):

\[ \sin \theta = \frac{x}{\sqrt{x^2 + y^2}} \quad \text{and} \quad \cos \theta = \frac{y}{\sqrt{x^2 + y^2}}. \]

### Step 3: Substitute \( \sin \theta \) and \( \cos \theta \) into the expression
Now, substitute \( \sin \theta \) and \( \cos \theta \) into the expression:

\[ x \sin \theta + y \cos \theta = x \left(\frac{x}{\sqrt{x^2 + y^2}}\right) + y \left(\frac{y}{\sqrt{x^2 + y^2}}\right). \]

This simplifies to:

\[ \frac{x^2 + y^2}{\sqrt{x^2 + y^2}}. \]

Similarly, for the denominator:

\[ x \sin \theta - y \cos \theta = x \left(\frac{x}{\sqrt{x^2 + y^2}}\right) - y \left(\frac{y}{\sqrt{x^2 + y^2}}\right). \]

This simplifies to:

\[ \frac{x^2 - y^2}{\sqrt{x^2 + y^2}}. \]

### Step 4: Combine the results
Now we can combine the results into the original expression:

\[ \frac{x \sin \theta + y \cos \theta}{x \sin \theta - y \cos \theta} = \frac{\frac{x^2 + y^2}{\sqrt{x^2 + y^2}}}{\frac{x^2 - y^2}{\sqrt{x^2 + y^2}}}. \]

### Step 5: Simplify the expression
The \( \sqrt{x^2 + y^2} \) cancels out:

\[ = \frac{x^2 + y^2}{x^2 - y^2}. \]

Thus, the final result is:

\[ \frac{x^2 + y^2}{x^2 - y^2}. \]

### Final Answer
The expression \( \frac{x \sin \theta + y \cos \theta}{x \sin \theta - y \cos \theta} \) is equal to

\[ \frac{x^2 + y^2}{x^2 - y^2}. \]

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