`cosecθ//(cosecθ + 1) + cosecθ//(cosecθ - 1) - 2tan^(2)θ`

Options

A. `1//2`
B. 0
C. `-2`
D. 2

Answer

To solve the expression \( \frac{\csc \theta}{\csc \theta + 1} + \frac{\csc \theta}{\csc \theta - 1} - 2 \tan^2 \theta \), we will break it down step by step.

### Step 1: Rewrite Cosecant and Tangent
Recall that:
- \( \csc \theta = \frac{1}{\sin \theta} \)
- \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

Thus, \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \).

### Step 2: Substitute Cosecant
Substituting \( \csc \theta \) into the expression, we have:
\[ \frac{\frac{1}{\sin \theta}}{\frac{1}{\sin \theta} + 1} + \frac{\frac{1}{\sin \theta}}{\frac{1}{\sin \theta} - 1} - 2 \frac{\sin^2 \theta}{\cos^2 \theta} \]

### Step 3: Simplify the First Two Terms
Let's simplify the first term:
\[ \frac{\frac{1}{\sin \theta}}{\frac{1}{\sin \theta} + 1} = \frac{1}{1 + \sin \theta} \]
And the second term:
\[ \frac{\frac{1}{\sin \theta}}{\frac{1}{\sin \theta} - 1} = \frac{1}{1 - \sin \theta} \]

So, we can rewrite the expression as:
\[ \frac{1}{1 + \sin \theta} + \frac{1}{1 - \sin \theta} - 2 \frac{\sin^2 \theta}{\cos^2 \theta} \]

### Step 4: Combine the First Two Terms
To combine the first two fractions:
\[ \frac{1}{1 + \sin \theta} + \frac{1}{1 - \sin \theta} = \frac{(1 - \sin \theta) + (1 + \sin \theta)}{(1 + \sin \theta)(1 - \sin \theta)} = \frac{2}{1 - \sin^2 \theta} = \frac{2}{\cos^2 \theta} \]

### Step 5: Substitute Back
Now, substituting back into our expression gives:
\[ \frac{2}{\cos^2 \theta} - 2 \frac{\sin^2 \theta}{\cos^2 \theta} \]

### Step 6: Factor Out Common Terms
Factoring out \( \frac{2}{\cos^2 \theta} \):
\[ \frac{2}{\cos^2 \theta} (1 - \sin^2 \theta) = \frac{2}{\cos^2 \theta} \cos^2 \theta = 2 \]

### Final Answer
Thus, the final answer is:
\[ \boxed{2} \]

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