If `tan theta + sec theta = (x - 2)//(x + 2)`, then what is the value of cos `theta` ?

Options

A. `(x^(2)-1)//(x^(2)+1)`
B. `(2x^(2)-4)//(2x^(2)+ 4)`
C. `(x^(2)-4)//(x^(2)+4)`
D. `(x^(2)-2)//(x^(2)+2)`

Answer

To solve the equation \( \tan \theta + \sec \theta = \frac{x - 2}{x + 2} \) and find the value of \( \cos \theta \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \tan \theta + \sec \theta = \frac{x - 2}{x + 2} \]

### Step 2: Use the Pythagorean identity
Recall the Pythagorean identity:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
This can be rearranged to:
\[ \sec^2 \theta = 1 + \tan^2 \theta \]

### Step 3: Multiply both sides by \( \sec \theta - \tan \theta \)
We can multiply both sides of the original equation by \( \sec \theta - \tan \theta \):
\[ (\tan \theta + \sec \theta)(\sec \theta - \tan \theta) = \frac{x - 2}{x + 2}(\sec \theta - \tan \theta) \]
This simplifies to:
\[ \sec^2 \theta - \tan^2 \theta = \frac{x - 2}{x + 2}(\sec \theta - \tan \theta) \]
Using the identity we mentioned, we substitute:
\[ 1 = \frac{x - 2}{x + 2}(\sec \theta - \tan \theta) \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ \sec \theta - \tan \theta = \frac{x + 2}{x - 2} \]

### Step 5: Set up two equations
Now we have two equations:
1. \( \tan \theta + \sec \theta = \frac{x - 2}{x + 2} \)
2. \( \sec \theta - \tan \theta = \frac{x + 2}{x - 2} \)

### Step 6: Add the two equations
Adding these two equations:
\[ (\tan \theta + \sec \theta) + (\sec \theta - \tan \theta) = \frac{x - 2}{x + 2} + \frac{x + 2}{x - 2} \]
This simplifies to:
\[ 2 \sec \theta = \frac{x - 2}{x + 2} + \frac{x + 2}{x - 2} \]

### Step 7: Simplify the right side
To simplify the right side, we need a common denominator:
\[ \frac{(x - 2)^2 + (x + 2)^2}{(x + 2)(x - 2)} \]
Calculating the numerator:
\[ (x - 2)^2 + (x + 2)^2 = x^2 - 4x + 4 + x^2 + 4x + 4 = 2x^2 + 8 \]
So we have:
\[ 2 \sec \theta = \frac{2x^2 + 8}{x^2 - 4} \]

### Step 8: Solve for \( \sec \theta \)
Dividing both sides by 2:
\[ \sec \theta = \frac{x^2 + 4}{x^2 - 4} \]

### Step 9: Find \( \cos \theta \)
Since \( \sec \theta = \frac{1}{\cos \theta} \), we can find \( \cos \theta \):
\[ \cos \theta = \frac{x^2 - 4}{x^2 + 4} \]

### Final Answer
Thus, the value of \( \cos \theta \) is:
\[ \cos \theta = \frac{x^2 - 4}{x^2 + 4} \]