The value of `theta [0^(@) lt theta lt 90^(@)]`, for which `(cos theta)/(1-sin theta ) + (cos theta)/(1+ sin theta) =4`, is:

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. None of these

Answer

To solve the equation

\[ \frac{\cos \theta}{1 - \sin \theta} + \frac{\cos \theta}{1 + \sin \theta} = 4, \]

we will follow these steps:

### Step 1: Combine the fractions
We can combine the two fractions on the left side by finding a common denominator. The common denominator is \((1 - \sin \theta)(1 + \sin \theta)\).

\[ \frac{\cos \theta(1 + \sin \theta) + \cos \theta(1 - \sin \theta)}{(1 - \sin \theta)(1 + \sin \theta)} = 4 \]

### 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:

\[ \frac{2 \cos \theta}{(1 - \sin \theta)(1 + \sin \theta)} = 4 \]

### 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:

\[ \frac{2 \cos \theta}{\cos^2 \theta} = 4 \]

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

\[ \frac{2}{\cos \theta} = 4 \]

### Step 6: Solve for \(\cos \theta\)
Now, we can solve for \(\cos \theta\):

\[ 2 = 4 \cos \theta \implies \cos \theta = \frac{1}{2} \]

### Step 7: Find \(\theta\)
Now, we need to find \(\theta\) such that \(0^\circ < \theta < 90^\circ\):

\[ \theta = \cos^{-1}\left(\frac{1}{2}\right) = 60^\circ \]

Thus, the value of \(\theta\) is:

\[ \boxed{60^\circ} \]