If `(costheta)/(1+sintheta)+(costheta)/(1-sintheta)=2sqrt(2)` and `theta` is acute , then what is the value (in degrees ) of `theta` ?

Options

A. 30
B. 45
C. 60
D. 90

Answer

To solve the equation

\[ \frac{\cos \theta}{1 + \sin \theta} + \frac{\cos \theta}{1 - \sin \theta} = 2\sqrt{2} \]

we will follow these steps:

### Step 1: Combine the fractions
We need to find a common denominator for the two fractions on the left side. The common denominator is \((1 + \sin \theta)(1 - \sin \theta)\).

\[ \frac{\cos \theta (1 - \sin \theta) + \cos \theta (1 + \sin \theta)}{(1 + \sin \theta)(1 - \sin \theta)} = 2\sqrt{2} \]

### Step 2: Simplify the numerator
Now, simplify the numerator:

\[ \cos \theta (1 - \sin \theta) + \cos \theta (1 + \sin \theta) = \cos \theta - \cos \theta \sin \theta + \cos \theta + \cos \theta \sin \theta = 2\cos \theta \]

So, we have:

\[ \frac{2\cos \theta}{(1 + \sin \theta)(1 - \sin \theta)} = 2\sqrt{2} \]

### Step 3: Simplify the denominator
The denominator can be simplified using the identity \(1 - \sin^2 \theta = \cos^2 \theta\):

\[ (1 + \sin \theta)(1 - \sin \theta) = 1 - \sin^2 \theta = \cos^2 \theta \]

### Step 4: Substitute back into the equation
Now substitute this back into the equation:

\[ \frac{2\cos \theta}{\cos^2 \theta} = 2\sqrt{2} \]

### Step 5: Simplify the equation
This simplifies to:

\[ \frac{2}{\cos \theta} = 2\sqrt{2} \]

### Step 6: Solve for \(\cos \theta\)
Dividing both sides by 2 gives:

\[ \frac{1}{\cos \theta} = \sqrt{2} \]

Taking the reciprocal gives:

\[ \cos \theta = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]

### Step 7: Find \(\theta\)
The angle \(\theta\) that satisfies \(\cos \theta = \frac{\sqrt{2}}{2}\) in the acute range is:

\[ \theta = 45^\circ \]

### Final Answer
Thus, the value of \(\theta\) is

\[ \boxed{45} \]