Which of the following is not equal to: `(tan theta + sectheta-1)/(tan theta - sectheta +1)`

Options

A. `sectheta + tan theta`
B. `1/(sectheta + tan theta)`
C. `(1+ sin theta)/(cos theta)`
D. `(1- sin theta)/(cos theta)`

Answer

To solve the problem, we need to evaluate the expression \((\tan \theta + \sec \theta - 1) / (\tan \theta - \sec \theta + 1)\) and determine which of the given options is not equal to this expression.

### Step-by-step Solution:

1. **Substituting \(\theta = 45^\circ\)**:
- We know that \(\tan 45^\circ = 1\) and \(\sec 45^\circ = \sqrt{2}\).
- Substitute these values into the expression:
\[ \text{Numerator: } \tan 45^\circ + \sec 45^\circ - 1 = 1 + \sqrt{2} - 1 = \sqrt{2} \]
\[ \text{Denominator: } \tan 45^\circ - \sec 45^\circ + 1 = 1 - \sqrt{2} + 1 = 2 - \sqrt{2} \]

2. **Forming the Expression**:
- Now we can form the expression:
\[ \frac{\tan 45^\circ + \sec 45^\circ - 1}{\tan 45^\circ - \sec 45^\circ + 1} = \frac{\sqrt{2}}{2 - \sqrt{2}} \]

3. **Rationalizing the Denominator**:
- To simplify, we multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{\sqrt{2}}{2 - \sqrt{2}} \cdot \frac{2 + \sqrt{2}}{2 + \sqrt{2}} = \frac{\sqrt{2}(2 + \sqrt{2})}{(2 - \sqrt{2})(2 + \sqrt{2})} \]
- The denominator simplifies to:
\[ (2 - \sqrt{2})(2 + \sqrt{2}) = 4 - 2 = 2 \]
- The numerator simplifies to:
\[ \sqrt{2}(2 + \sqrt{2}) = 2\sqrt{2} + 2 \]

4. **Final Expression**:
- Therefore, the entire expression simplifies to:
\[ \frac{2\sqrt{2} + 2}{2} = \sqrt{2} + 1 \]

5. **Checking Options**:
- We need to check which of the options does not equal \(\sqrt{2} + 1\):
- **Option 1**: \(\sec \theta + \tan \theta\) at \(\theta = 45^\circ\) gives \(\sqrt{2} + 1\).
- **Option 2**: \(\frac{1}{\sec \theta + \tan \theta}\) gives \(\frac{1}{\sqrt{2} + 1}\), which does not equal \(\sqrt{2} + 1\).
- **Option 3**: \(\frac{1 + \sin \theta}{\cos \theta}\) gives \(\sqrt{2} + 1\).
- **Option 4**: \(\frac{1 - \sin \theta}{\cos \theta}\) gives \(\sqrt{2} - 1\), which does not equal \(\sqrt{2} + 1\).

### Conclusion:
The options that are not equal to \(\sqrt{2} + 1\) are Option 2 and Option 4.