`((1+costheta)^(2)+sin^(2)theta)/((cosec^(2)theta-1)sin^(2)theta)=`

Options

A. `cos theta (1 + sin theta)`
B. `sec theta(1 + sin theta)`
C. `2 cos theta(1 + sec theta)`
D. `2 sec theta (1 + sec theta)`

Answer

To solve the expression \(\frac{(1 + \cos \theta)^2 + \sin^2 \theta}{(\csc^2 \theta - 1) \sin^2 \theta}\), we will simplify both the numerator and the denominator step by step.

### Step 1: Simplify the Numerator
The numerator is \((1 + \cos \theta)^2 + \sin^2 \theta\).

1. Expand \((1 + \cos \theta)^2\):
\[ (1 + \cos \theta)^2 = 1 + 2\cos \theta + \cos^2 \theta \]

2. Now, add \(\sin^2 \theta\):
\[ 1 + 2\cos \theta + \cos^2 \theta + \sin^2 \theta \]

3. Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\):
\[ = 1 + 2\cos \theta + 1 = 2 + 2\cos \theta \]

So, the numerator simplifies to:
\[ 2(1 + \cos \theta) \]

### Step 2: Simplify the Denominator
The denominator is \((\csc^2 \theta - 1) \sin^2 \theta\).

1. Recall that \(\csc^2 \theta = \frac{1}{\sin^2 \theta}\), so:
\[ \csc^2 \theta - 1 = \frac{1}{\sin^2 \theta} - 1 = \frac{1 - \sin^2 \theta}{\sin^2 \theta} \]

2. Using the Pythagorean identity again, \(1 - \sin^2 \theta = \cos^2 \theta\):
\[ \csc^2 \theta - 1 = \frac{\cos^2 \theta}{\sin^2 \theta} \]

3. Now, multiply by \(\sin^2 \theta\):
\[ (\csc^2 \theta - 1) \sin^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} \cdot \sin^2 \theta = \cos^2 \theta \]

### Step 3: Combine the Results
Now we can substitute the simplified numerator and denominator back into the expression:
\[ \frac{2(1 + \cos \theta)}{\cos^2 \theta} \]

### Step 4: Final Simplification
This can be further simplified:
\[ = \frac{2(1 + \cos \theta)}{\cos^2 \theta} = 2 \cdot \frac{1 + \cos \theta}{\cos^2 \theta} \]

### Final Answer
Thus, the final answer is:
\[ \frac{2(1 + \cos \theta)}{\cos^2 \theta} \]

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