`(1+tan theta+sec theta)(1+cot theta-"cosec "theta)` is equal to :

Options

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

Answer

To solve the expression \((1 + \tan \theta + \sec \theta)(1 + \cot \theta - \csc \theta)\), we will follow these steps:

### Step 1: Rewrite the trigonometric functions in terms of sine and cosine
We know that:
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
- \(\sec \theta = \frac{1}{\cos \theta}\)
- \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)
- \(\csc \theta = \frac{1}{\sin \theta}\)

So, we rewrite the expression:
\[ (1 + \frac{\sin \theta}{\cos \theta} + \frac{1}{\cos \theta})(1 + \frac{\cos \theta}{\sin \theta} - \frac{1}{\sin \theta}) \]

### Step 2: Simplify each bracket
The first bracket becomes:
\[ 1 + \frac{\sin \theta + 1}{\cos \theta} = \frac{\cos \theta + \sin \theta + 1}{\cos \theta} \]

The second bracket becomes:
\[ 1 + \frac{\cos \theta - 1}{\sin \theta} = \frac{\sin \theta + \cos \theta - 1}{\sin \theta} \]

### Step 3: Multiply the two simplified brackets
Now we multiply the two fractions:
\[ \frac{(\cos \theta + \sin \theta + 1)(\sin \theta + \cos \theta - 1)}{\cos \theta \sin \theta} \]

### Step 4: Expand the numerator
Using the distributive property (FOIL method), we expand:
\[ (\cos \theta + \sin \theta + 1)(\sin \theta + \cos \theta - 1) = (\cos \theta + \sin \theta)(\sin \theta + \cos \theta) + (\cos \theta + \sin \theta)(-1) \]
This simplifies to:
\[ (\cos^2 \theta + \sin^2 \theta + 2\sin \theta \cos \theta) - (\cos \theta + \sin \theta) \]

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

### Step 6: Combine like terms
The numerator now is:
\[ (1 - 1) + 2\sin \theta \cos \theta - \cos \theta - \sin \theta = 2\sin \theta \cos \theta - \cos \theta - \sin \theta \]

### Step 7: Factor the numerator
We can factor out \(\sin \theta + \cos \theta\):
\[ = (2\sin \theta \cos \theta - (\sin \theta + \cos \theta)) \]

### Step 8: Divide by the denominator
Now we divide by the denominator:
\[ \frac{2\sin \theta \cos \theta - (\sin \theta + \cos \theta)}{\cos \theta \sin \theta} \]

### Step 9: Simplify the fraction
This simplifies to:
\[ \frac{2\sin \theta \cos \theta}{\cos \theta \sin \theta} - \frac{\sin \theta + \cos \theta}{\cos \theta \sin \theta} \]
The first term simplifies to \(2\) and the second term simplifies to \(1\).

### Final Result
Thus, the final result is:
\[ 2 \]