If `(1/(1+ "cosec" theta)-1/(1-"cosec" theta))costheta =2,0^@ lt theta lt 90^@` then the value of `sin^2 theta + cot^2 theta + sec^2 theta` is :

Options

A. 1
B. `2 1/2`
C. `3 1/2`
D. 2

Answer

To solve the problem step by step, we start with the given equation:

\[ \left( \frac{1}{1 + \csc \theta} - \frac{1}{1 - \csc \theta} \right) \cos \theta = 2 \]

### Step 1: Simplify the left-hand side
We first find a common denominator for the fractions:

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

This simplifies to:

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

Using the identity \( \csc^2 \theta = 1 + \cot^2 \theta \), we rewrite the denominator:

\[ 1 - \csc^2 \theta = -\cot^2 \theta \]

Thus, the expression becomes:

\[ \frac{-2 \csc \theta}{-\cot^2 \theta} = \frac{2 \csc \theta}{\cot^2 \theta} \]

### Step 2: Substitute back into the equation
Now substituting back into the equation gives us:

\[ \frac{2 \csc \theta}{\cot^2 \theta} \cos \theta = 2 \]

Since \( \csc \theta = \frac{1}{\sin \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), we can rewrite this as:

\[ \frac{2 \cdot \frac{1}{\sin \theta}}{\left(\frac{\cos \theta}{\sin \theta}\right)^2} \cos \theta = 2 \]

This simplifies to:

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

### Step 3: Further simplification
This leads to:

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

Dividing both sides by 2 gives:

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

### Step 4: Solve for \(\theta\)
This can be rewritten as:

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

Using the identity \( \sin \theta = 1 - \cos^2 \theta \):

\[ \cos^2 \theta = 1 - \cos^2 \theta \]

This leads to:

\[ 2 \cos^2 \theta = 1 \implies \cos^2 \theta = \frac{1}{2} \implies \cos \theta = \frac{1}{\sqrt{2}} \implies \theta = 45^\circ \]

### Step 5: Calculate \( \sin^2 \theta + \cot^2 \theta + \sec^2 \theta \)
Now we need to find:

\[ \sin^2 \theta + \cot^2 \theta + \sec^2 \theta \]

Calculating each term for \( \theta = 45^\circ \):

- \( \sin^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \)
- \( \cot^2 45^\circ = 1^2 = 1 \)
- \( \sec^2 45^\circ = \left(\frac{1}{\cos 45^\circ}\right)^2 = \left(\sqrt{2}\right)^2 = 2 \)

Adding these together:

\[ \sin^2 \theta + \cot^2 \theta + \sec^2 \theta = \frac{1}{2} + 1 + 2 = \frac{1}{2} + \frac{2}{2} + \frac{4}{2} = \frac{7}{2} \]

### Final Answer
Thus, the value of \( \sin^2 \theta + \cot^2 \theta + \sec^2 \theta \) is:

\[ \frac{7}{2} \]