If `(cos theta +sin theta) : ( cos theta - sin theta) = (sqrt(3) +1): (sqrt(3)-1), 0^(@) lt theta lt 90^(@)` , then what is the value of `sec theta` ?

Options

A. 1
B. 2
C. `sqrt(2)`
D. `{:(2+sqrt(3)),(" "3):}`

Answer

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

\[ \frac{\cos \theta + \sin \theta}{\cos \theta - \sin \theta} = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \]

### Step 1: Cross Multiply
We will cross-multiply to eliminate the fractions:

\[ (\cos \theta + \sin \theta)(\sqrt{3} - 1) = (\cos \theta - \sin \theta)(\sqrt{3} + 1) \]

### Step 2: Expand Both Sides
Now, we expand both sides:

\[ \sqrt{3} \cos \theta - \cos \theta + \sqrt{3} \sin \theta - \sin \theta = \sqrt{3} \cos \theta + \cos \theta - \sqrt{3} \sin \theta - \sin \theta \]

### Step 3: Rearrange the Equation
Next, we will rearrange the equation to group similar terms:

\[ \sqrt{3} \cos \theta + \sqrt{3} \sin \theta - \cos \theta - \sin \theta = \sqrt{3} \cos \theta + \cos \theta - \sqrt{3} \sin \theta - \sin \theta \]

This simplifies to:

\[ \sqrt{3} \cos \theta + \sqrt{3} \sin \theta - \sqrt{3} \cos \theta - \sqrt{3} \sin \theta = 2 \cos \theta - 2 \sin \theta \]

### Step 4: Combine Like Terms
Combining like terms gives us:

\[ 0 = 2 \cos \theta - 2 \sin \theta \]

### Step 5: Factor Out Common Terms
Factoring out the common terms:

\[ 2(\cos \theta - \sin \theta) = 0 \]

### Step 6: Solve for Cosine and Sine
This implies:

\[ \cos \theta = \sin \theta \]

### Step 7: Find the Value of Theta
The solution to \(\cos \theta = \sin \theta\) in the interval \(0^\circ < \theta < 90^\circ\) is:

\[ \theta = 45^\circ \]

### Step 8: Find Secant of Theta
Now, we need to find the value of \(\sec \theta\):

\[ \sec \theta = \frac{1}{\cos \theta} \]

Since \(\theta = 45^\circ\):

\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \]

Thus:

\[ \sec 45^\circ = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]

### Final Answer
The value of \(\sec \theta\) is:

\[ \sqrt{2} \]