if `(cos theta)/(1-sin theta)+ (cos theta)/(1 + sin theta) = 4 , 0^@ lt theta lt 90^@` then what is the value of `(sec theta + cosec theta + cot theta)?`

Options

A. `1 + 2 sqrt(3)`
B. `(1+2 sqrt(3))/(3)`
C. `(2 + sqrt(3))/(3)`
D. `2 + sqrt(3)`

Answer

To solve the equation

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

we will start by simplifying the left-hand side.

### Step 1: Combine the fractions

The left-hand side can be combined into a single fraction:

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

### Step 2: Simplify the numerator

Now, simplify the numerator:

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

### Step 3: Simplify the denominator

The denominator simplifies as follows:

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

### Step 4: Rewrite the equation

Now we can rewrite 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

Since \(0^\circ < \theta < 90^\circ\), we find:

\[ \theta = 60^\circ \]

### Step 8: Calculate sec theta, cosec theta, and cot theta

Now we need to find the value of:

\[ \sec \theta + \csc \theta + \cot \theta \]

Calculating each term:

- \(\sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{1}{2}} = 2\)
- \(\csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}\)
- \(\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}\)

### Step 9: Combine the results

Now, we combine these results:

\[ \sec \theta + \csc \theta + \cot \theta = 2 + \frac{2}{\sqrt{3}} + \frac{1}{\sqrt{3}} = 2 + \frac{3}{\sqrt{3}} = 2 + \sqrt{3} \]

### Final Answer

Thus, the value of \(\sec \theta + \csc \theta + \cot \theta\) is:

\[ \boxed{2 + \sqrt{3}} \]

---