What is the value of `((cottheta+cosec theta-1))/((cottheta-cosectheta+1))`?

Options

A. `cot theta+cosec theta`
B. 1
C. `-1`
D. 0

Answer

To solve the expression \(\frac{\cot \theta + \csc \theta - 1}{\cot \theta - \csc \theta + 1}\), we can substitute \(\theta = 45^\circ\) to find its value.

### Step-by-Step Solution:

1. **Substitute \(\theta = 45^\circ\)**:
\[ \cot 45^\circ = 1 \quad \text{and} \quad \csc 45^\circ = \sqrt{2} \]

2. **Calculate the numerator**:
\[ \cot 45^\circ + \csc 45^\circ - 1 = 1 + \sqrt{2} - 1 = \sqrt{2} \]

3. **Calculate the denominator**:
\[ \cot 45^\circ - \csc 45^\circ + 1 = 1 - \sqrt{2} + 1 = 2 - \sqrt{2} \]

4. **Form the expression**:
\[ \frac{\sqrt{2}}{2 - \sqrt{2}} \]

5. **Rationalize the denominator**:
Multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{\sqrt{2} \cdot (2 + \sqrt{2})}{(2 - \sqrt{2})(2 + \sqrt{2})} \]

6. **Calculate the denominator**:
\[ (2 - \sqrt{2})(2 + \sqrt{2}) = 4 - 2 = 2 \]

7. **Calculate the numerator**:
\[ \sqrt{2} \cdot (2 + \sqrt{2}) = 2\sqrt{2} + 2 \]

8. **Combine the results**:
\[ \frac{2\sqrt{2} + 2}{2} = \sqrt{2} + 1 \]

### Final Result:
Thus, the value of \(\frac{\cot \theta + \csc \theta - 1}{\cot \theta - \csc \theta + 1}\) when \(\theta = 45^\circ\) is:
\[ \sqrt{2} + 1 \]