If `tan theta + sec theta = m`, then what is `sec theta` equal to ?

Options

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

Answer

To find the value of `sec theta` in terms of `m`, given that `tan theta + sec theta = m`, we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ \tan \theta + \sec \theta = m \]

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

### Step 3: Express `tan theta` in terms of `sec theta`
From the original equation, we can express `tan theta` as:
\[ \tan \theta = m - \sec \theta \]

### Step 4: Substitute `tan theta` into the Pythagorean identity
Substituting `tan theta` into the identity gives:
\[ \sec^2 \theta = 1 + (m - \sec \theta)^2 \]

### Step 5: Expand the equation
Expanding the right side:
\[ \sec^2 \theta = 1 + (m^2 - 2m \sec \theta + \sec^2 \theta) \]
This simplifies to:
\[ \sec^2 \theta = 1 + m^2 - 2m \sec \theta + \sec^2 \theta \]

### Step 6: Cancel `sec^2 theta` from both sides
Subtract `sec^2 theta` from both sides:
\[ 0 = 1 + m^2 - 2m \sec \theta \]

### Step 7: Rearrange to solve for `sec theta`
Rearranging gives:
\[ 2m \sec \theta = 1 + m^2 \]
Thus,
\[ \sec \theta = \frac{1 + m^2}{2m} \]

### Final Result
The value of `sec theta` in terms of `m` is:
\[ \sec \theta = \frac{1 + m^2}{2m} \]