If `(sectheta - tantheta)/(sectheta + tantheta) = 3/5`,then the value of `("cosec"theta + cot theta)/("cosec"theta - cottheta)` is:

Options

A. `27 + sqrt(15)`
B. `33 + 4sqrt(15)`
C. `31 + 8sqrt(15)`
D. `24 + sqrt(15)`

Answer

To solve the equation \(\frac{\sec \theta - \tan \theta}{\sec \theta + \tan \theta} = \frac{3}{5}\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \frac{\sec \theta - \tan \theta}{\sec \theta + \tan \theta} = \frac{3}{5} \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ 5(\sec \theta - \tan \theta) = 3(\sec \theta + \tan \theta) \]

### Step 3: Expand both sides
Expanding both sides:
\[ 5 \sec \theta - 5 \tan \theta = 3 \sec \theta + 3 \tan \theta \]

### Step 4: Rearranging the equation
Rearranging the terms:
\[ 5 \sec \theta - 3 \sec \theta = 5 \tan \theta + 3 \tan \theta \]
This simplifies to:
\[ 2 \sec \theta = 8 \tan \theta \]

### Step 5: Simplifying the equation
Dividing both sides by 2:
\[ \sec \theta = 4 \tan \theta \]

### Step 6: Using trigonometric identities
Recall that \(\sec \theta = \frac{1}{\cos \theta}\) and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substituting these into the equation gives:
\[ \frac{1}{\cos \theta} = 4 \cdot \frac{\sin \theta}{\cos \theta} \]
This simplifies to:
\[ 1 = 4 \sin \theta \]
Thus,
\[ \sin \theta = \frac{1}{4} \]

### Step 7: Finding \(\cos \theta\)
Using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\):
\[ \left(\frac{1}{4}\right)^2 + \cos^2 \theta = 1 \]
\[ \frac{1}{16} + \cos^2 \theta = 1 \]
\[ \cos^2 \theta = 1 - \frac{1}{16} = \frac{15}{16} \]
Thus,
\[ \cos \theta = \frac{\sqrt{15}}{4} \]

### Step 8: Finding \(\csc \theta\) and \(\cot \theta\)
Now we can find \(\csc \theta\) and \(\cot \theta\):
\[ \csc \theta = \frac{1}{\sin \theta} = 4 \]
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\frac{\sqrt{15}}{4}}{\frac{1}{4}} = \sqrt{15} \]

### Step 9: Calculate \(\frac{\csc \theta + \cot \theta}{\csc \theta - \cot \theta}\)
Now substituting these values into the expression:
\[ \frac{\csc \theta + \cot \theta}{\csc \theta - \cot \theta} = \frac{4 + \sqrt{15}}{4 - \sqrt{15}} \]

### Step 10: Rationalizing the denominator
To rationalize the denominator:
\[ \frac{(4 + \sqrt{15})(4 + \sqrt{15})}{(4 - \sqrt{15})(4 + \sqrt{15})} = \frac{(4 + \sqrt{15})^2}{16 - 15} \]
Calculating the numerator:
\[ (4 + \sqrt{15})^2 = 16 + 8\sqrt{15} + 15 = 31 + 8\sqrt{15} \]
Thus, we have:
\[ \frac{31 + 8\sqrt{15}}{1} = 31 + 8\sqrt{15} \]

### Final Answer
The value of \(\frac{\csc \theta + \cot \theta}{\csc \theta - \cot \theta}\) is:
\[ \boxed{31 + 8\sqrt{15}} \]