The identity `(1+ tan theta - sec theta)(1+ cot theta - "cosec"theta)` not equals

Options

A. 2
B. 1
C. `2 tan theta`
D. `2 cot theta`

Answer

To solve the identity \((1 + \tan \theta - \sec \theta)(1 + \cot \theta - \csc \theta)\) and show that it does not equal any of the given options, we can evaluate the expression by substituting a specific angle for \(\theta\). A common choice is \(\theta = 45^\circ\) since it simplifies the trigonometric functions involved.

### Step-by-step Solution:

1. **Substituting \(\theta = 45^\circ\)**:
\[ \tan 45^\circ = 1, \quad \sec 45^\circ = \sqrt{2}, \quad \cot 45^\circ = 1, \quad \csc 45^\circ = \sqrt{2} \]
Substitute these values into the expression:
\[ (1 + \tan 45^\circ - \sec 45^\circ)(1 + \cot 45^\circ - \csc 45^\circ) \]
becomes:
\[ (1 + 1 - \sqrt{2})(1 + 1 - \sqrt{2}) \]

2. **Simplifying the expression**:
\[ (2 - \sqrt{2})(2 - \sqrt{2}) \]
This can be rewritten as:
\[ (2 - \sqrt{2})^2 \]

3. **Expanding the squared term**:
\[ (2 - \sqrt{2})^2 = 2^2 - 2 \cdot 2 \cdot \sqrt{2} + (\sqrt{2})^2 \]
Simplifying this gives:
\[ 4 - 4\sqrt{2} + 2 = 6 - 4\sqrt{2} \]

4. **Final value**:
The value of the expression is \(6 - 4\sqrt{2}\).

### Conclusion:
Since \(6 - 4\sqrt{2}\) is not equal to \(2\), \(2\tan \theta\), or \(2\cot \theta\) when \(\theta = 45^\circ\), we can conclude that the identity does not equal any of the provided options.