If `2 cos theta = x + ( 1)/( x) , 2 cos phi = y + ( 1)/( y )`, then `cos ( theta - phi ) =`

Options

A. `(x)/( y ) + ( y )/( x)`
B. `xy + ( 1)/( xy )`
C. `(1)/( 2) ((x)/( y ) + ( y )/( x))`
D. none

Answer

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

1. \( 2 \cos \theta = x + \frac{1}{x} \)
2. \( 2 \cos \phi = y + \frac{1}{y} \)

We need to find \( \cos(\theta - \phi) \).

### Step 1: Express \( \cos \theta \) and \( \cos \phi \)

From the first equation, we can express \( \cos \theta \):
\[ \cos \theta = \frac{x + \frac{1}{x}}{2} \]

From the second equation, we can express \( \cos \phi \):
\[ \cos \phi = \frac{y + \frac{1}{y}}{2} \]

### Step 2: Use the cosine subtraction formula

The cosine subtraction formula states:
\[ \cos(\theta - \phi) = \cos \theta \cos \phi + \sin \theta \sin \phi \]

### Step 3: Find \( \sin \theta \) and \( \sin \phi \)

Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can find \( \sin \theta \) and \( \sin \phi \).

First, we calculate \( \sin^2 \theta \):
\[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(\frac{x + \frac{1}{x}}{2}\right)^2 \]
\[ = 1 - \frac{x^2 + 2 + \frac{1}{x^2}}{4} = \frac{4 - (x^2 + 2 + \frac{1}{x^2})}{4} = \frac{2 - x^2 - \frac{1}{x^2}}{4} \]

Thus,
\[ \sin \theta = \sqrt{\frac{2 - x^2 - \frac{1}{x^2}}{4}} = \frac{\sqrt{2 - x^2 - \frac{1}{x^2}}}{2} \]

Similarly, for \( \sin \phi \):
\[ \sin^2 \phi = 1 - \cos^2 \phi = 1 - \left(\frac{y + \frac{1}{y}}{2}\right)^2 \]
\[ = \frac{2 - y^2 - \frac{1}{y^2}}{4} \]

Thus,
\[ \sin \phi = \frac{\sqrt{2 - y^2 - \frac{1}{y^2}}}{2} \]

### Step 4: Substitute into the cosine subtraction formula

Now substituting \( \cos \theta \), \( \cos \phi \), \( \sin \theta \), and \( \sin \phi \) into the cosine subtraction formula:
\[ \cos(\theta - \phi) = \left(\frac{x + \frac{1}{x}}{2}\right) \left(\frac{y + \frac{1}{y}}{2}\right) + \left(\frac{\sqrt{2 - x^2 - \frac{1}{x^2}}}{2}\right) \left(\frac{\sqrt{2 - y^2 - \frac{1}{y^2}}}{2}\right) \]

### Step 5: Simplify the expression

This gives us:
\[ \cos(\theta - \phi) = \frac{(x + \frac{1}{x})(y + \frac{1}{y})}{4} + \frac{\sqrt{(2 - x^2 - \frac{1}{x^2})(2 - y^2 - \frac{1}{y^2})}}{4} \]

### Final Result

Thus, we can conclude that:
\[ \cos(\theta - \phi) = \frac{(x + \frac{1}{x})(y + \frac{1}{y}) + \sqrt{(2 - x^2 - \frac{1}{x^2})(2 - y^2 - \frac{1}{y^2})}}{4} \]