If `tantheta+sectheta=((x-2))/((x+2))` , then what is the value of `costheta` ?

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:
\[ \tan \theta + \sec \theta = \frac{x - 2}{x + 2} \]
We can express \( \tan \theta \) in terms of \( \sec \theta \):
\[ \sec \theta + \tan \theta = \frac{x - 2}{x + 2} \]

### Step 2: Multiply by the conjugate
Multiply both sides by the conjugate of the left-hand side:
\[ (\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = \frac{x - 2}{x + 2} \cdot (\sec \theta - \tan \theta) \]
This gives us:
\[ \sec^2 \theta - \tan^2 \theta = \frac{(x - 2)(\sec \theta - \tan \theta)}{x + 2} \]

### Step 3: Use the identity
Using the identity \( \sec^2 \theta - \tan^2 \theta = 1 \):
\[ 1 = \frac{(x - 2)(\sec \theta - \tan \theta)}{x + 2} \]

### Step 4: Cross-multiply
Cross-multiplying gives:
\[ x + 2 = (x - 2)(\sec \theta - \tan \theta) \]

### Step 5: Expand and rearrange
Expanding the right side:
\[ x + 2 = (x - 2)\sec \theta - (x - 2)\tan \theta \]
Rearranging gives:
\[ (x - 2)\sec \theta - (x - 2)\tan \theta - x - 2 = 0 \]

### Step 6: Solve for \( \sec \theta \)
Now we can isolate \( \sec \theta \):
\[ (x - 2)\sec \theta = (x - 2)\tan \theta + x + 2 \]
Thus,
\[ \sec \theta = \frac{(x - 2)\tan \theta + x + 2}{x - 2} \]

### Step 7: Find \( \cos \theta \)
Since \( \sec \theta = \frac{1}{\cos \theta} \), we can express \( \cos \theta \) as:
\[ \cos \theta = \frac{x - 2}{(x - 2)\tan \theta + x + 2} \]

### Step 8: Substitute \( \tan \theta \)
Using \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we can further simplify or substitute as needed.

### Final Result
After simplifying, we find:
\[ \cos \theta = \frac{x^2 - 4}{x^2 + 4} \]