If \(\frac{1}{{{\rm{cosec\theta \;}} + {\rm{\;}}1}}{\rm{\;}} + {\rm{\;}}\frac{1}{{{\rm{cosec\theta }} - 1}}{\rm{\;}} = {\rm{\;}}2{\rm{sec\theta }},0^\circ < {\rm{\theta }} < 90^\circ ,\) then the value of \(\frac{{{\rm{tan\theta \;}} + {\rm{\;}}2\sec {\rm{\theta }}}}{{{\rm{cosec\theta }}}}\) is

Options

A. `{:(2+sqrt(3))/(" "2):}`
B. `{:(4+sqrt(2))/(" "2):}`
C. `{:(2+sqrt(2))/(" "2):}`
D. `{:(4+sqrt(3))/(" "2):}`

Answer

To solve the equation

\[ \frac{1}{{\csc \theta + 1}} + \frac{1}{{\csc \theta - 1}} = 2 \sec \theta \]

we will follow these steps:

### Step 1: Rewrite the equation in terms of sine and cosine
Recall that \(\csc \theta = \frac{1}{\sin \theta}\) and \(\sec \theta = \frac{1}{\cos \theta}\). Therefore, we can rewrite the equation as:

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

### Step 2: Simplify the left-hand side
The left-hand side can be simplified by finding a common denominator:

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

Now, finding a common denominator for these two fractions:

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

### Step 3: Combine the fractions
This simplifies to:

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

### Step 4: Set the equation equal to the right-hand side
Now we can set this equal to the right-hand side:

\[ \frac{2 \sin \theta}{\cos^2 \theta} = 2 \cdot \frac{1}{\cos \theta} \]

### Step 5: Cross-multiply
Cross-multiplying gives us:

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

### Step 6: Simplify the equation
Dividing both sides by 2, we get:

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

### Step 7: Solve for \(\theta\)
This implies:

\[ \tan \theta = 1 \implies \theta = 45^\circ \]

### Step 8: Find the required expression
Now we need to find the value of

\[ \frac{\tan \theta + 2 \sec \theta}{\csc \theta} \]

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

- \(\tan 45^\circ = 1\)
- \(\sec 45^\circ = \sqrt{2}\)
- \(\csc 45^\circ = \sqrt{2}\)

Thus, we have:

\[ \frac{1 + 2\sqrt{2}}{\sqrt{2}} \]

### Step 9: Simplify the expression
This simplifies to:

\[ \frac{1}{\sqrt{2}} + 2 = \frac{1 + 2\sqrt{2}}{\sqrt{2}} = \frac{1 + 2\sqrt{2}}{\sqrt{2}} = \frac{1}{\sqrt{2}} + 2 \]

### Final Result
The final value is:

\[ \frac{1 + 2\sqrt{2}}{\sqrt{2}} = 2 + \frac{1}{\sqrt{2}} \approx 2 + 0.707 \approx 2.707 \]