If `x= sec theta + tan theta,` then `x+1/x=.`

Options

A. 1
B. `2 sec theta`
C. `2`
D. `2 tan theta`

Answer

To solve the problem where \( x = \sec \theta + \tan \theta \) and we need to find \( x + \frac{1}{x} \), we can follow these steps:

### Step 1: Write down the expression for \( x \)
Given:
\[ x = \sec \theta + \tan \theta \]

### Step 2: Find \( \frac{1}{x} \)
To find \( \frac{1}{x} \), we can use the identity:
\[ \frac{1}{x} = \frac{1}{\sec \theta + \tan \theta} \]
We can multiply the numerator and denominator by \( \sec \theta - \tan \theta \) (the conjugate of the denominator):
\[ \frac{1}{x} = \frac{\sec \theta - \tan \theta}{(\sec \theta + \tan \theta)(\sec \theta - \tan \theta)} = \frac{\sec \theta - \tan \theta}{\sec^2 \theta - \tan^2 \theta} \]
Using the identity \( \sec^2 \theta - \tan^2 \theta = 1 \):
\[ \frac{1}{x} = \sec \theta - \tan \theta \]

### Step 3: Add \( x \) and \( \frac{1}{x} \)
Now we can find \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = (\sec \theta + \tan \theta) + (\sec \theta - \tan \theta) \]
The \( \tan \theta \) terms cancel out:
\[ x + \frac{1}{x} = 2 \sec \theta \]

### Final Result
Thus, we have:
\[ x + \frac{1}{x} = 2 \sec \theta \]