`(1+cottheta-cosectheta)(1+tantheta+sectheta)=?`

Options

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

Answer

To solve the expression \((1 + \cot \theta - \csc \theta)(1 + \tan \theta + \sec \theta)\), we will simplify each part step by step.

### Step 1: Rewrite the trigonometric functions
We know the following identities:
- \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)
- \(\csc \theta = \frac{1}{\sin \theta}\)
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
- \(\sec \theta = \frac{1}{\cos \theta}\)

Substituting these identities into the expression, we have:
\[ (1 + \frac{\cos \theta}{\sin \theta} - \frac{1}{\sin \theta})(1 + \frac{\sin \theta}{\cos \theta} + \frac{1}{\cos \theta}) \]

### Step 2: Simplify the first bracket
The first bracket becomes:
\[ 1 + \frac{\cos \theta - 1}{\sin \theta} \]
Combining the terms:
\[ \frac{\sin \theta + \cos \theta - 1}{\sin \theta} \]

### Step 3: Simplify the second bracket
The second bracket becomes:
\[ 1 + \frac{\sin \theta + 1}{\cos \theta} \]
Combining the terms:
\[ \frac{\cos \theta + \sin \theta + 1}{\cos \theta} \]

### Step 4: Multiply the two brackets
Now we multiply the two simplified brackets:
\[ \left(\frac{\sin \theta + \cos \theta - 1}{\sin \theta}\right) \left(\frac{\cos \theta + \sin \theta + 1}{\cos \theta}\right) \]
This results in:
\[ \frac{(\sin \theta + \cos \theta - 1)(\cos \theta + \sin \theta + 1)}{\sin \theta \cos \theta} \]

### Step 5: Expand the numerator
Expanding the numerator:
\[ (\sin \theta + \cos \theta - 1)(\sin \theta + \cos \theta + 1) = (\sin^2 \theta + 2\sin \theta \cos \theta + \cos^2 \theta - 1) \]
Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\):
\[ = 1 + 2\sin \theta \cos \theta - 1 = 2\sin \theta \cos \theta \]

### Step 6: Final expression
Thus, the expression simplifies to:
\[ \frac{2\sin \theta \cos \theta}{\sin \theta \cos \theta} = 2 \]

### Final Answer:
\[ \boxed{2} \]