The value of ` (1 + cot theta - cosec theta ) (1 + tan theta +sec theta)` is

Options

A. 1
B. 2
C. 4
D. 0

Explanation

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 start by rewriting the cotangent, cosecant, tangent, and secant in terms of sine and cosine:
- \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)
- \(\csc \theta = \frac{1}{\sin \theta}\)
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
- \(\sec \theta = \frac{1}{\cos \theta}\)

Substituting these into the expression gives us:
\[ (1 + \frac{\cos \theta}{\sin \theta} - \frac{1}{\sin \theta})(1 + \frac{\sin \theta}{\cos \theta} + \frac{1}{\cos \theta}) \]

### Step 2: Simplify the first bracket
In the first bracket, we need a common denominator:
\[ 1 + \frac{\cos \theta - 1}{\sin \theta} = \frac{\sin \theta + \cos \theta - 1}{\sin \theta} \]

### Step 3: Simplify the second bracket
For the second bracket, we also need a common denominator:
\[ 1 + \frac{\sin \theta + 1}{\cos \theta} = \frac{\cos \theta + \sin \theta + 1}{\cos \theta} \]

### Step 4: Combine the two brackets
Now we can write the entire expression as:
\[ \frac{\sin \theta + \cos \theta - 1}{\sin \theta} \cdot \frac{\cos \theta + \sin \theta + 1}{\cos \theta} \]

### Step 5: Multiply the fractions
Multiplying these fractions gives:
\[ \frac{(\sin \theta + \cos \theta - 1)(\cos \theta + \sin \theta + 1)}{\sin \theta \cos \theta} \]

### Step 6: Expand the numerator
Now, we expand the numerator:
\[ (\sin \theta + \cos \theta - 1)(\sin \theta + \cos \theta + 1) = (\sin \theta + \cos \theta)^2 - 1^2 \]
Using the identity \(a^2 - b^2 = (a-b)(a+b)\):
\[ = \sin^2 \theta + \cos^2 \theta + 2\sin \theta \cos \theta - 1 \]
Since \(\sin^2 \theta + \cos^2 \theta = 1\), we have:
\[ = 1 + 2\sin \theta \cos \theta - 1 = 2\sin \theta \cos \theta \]

### Step 7: Final expression
Now substituting back into our expression, we get:
\[ \frac{2\sin \theta \cos \theta}{\sin \theta \cos \theta} = 2 \]

### Conclusion
Thus, the value of the expression \( (1 + \cot \theta - \csc \theta)(1 + \tan \theta + \sec \theta) \) is:
\[ \boxed{2} \]