`(sintheta-costheta+1)/(sintheta+costheta-1)` (where `theta ne (pi)/(2))` is equal to

Options

A. `(1+sintheta)/(costheta)`
B. `(1-sintheta)/(costheta)`
C. `(1-costheta)/(sintheta)`
D. `(1+costheta)/(sintheta)`

Answer

To solve the expression \(\frac{\sin \theta - \cos \theta + 1}{\sin \theta + \cos \theta - 1}\), we can follow these steps:

### Step 1: Substitute values for \(\sin \theta\) and \(\cos \theta\)
Let's substitute \(\theta = 30^\circ\) into the expression. We know that:
\[ \sin 30^\circ = \frac{1}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2} \]

### Step 2: Substitute into the expression
Now, we substitute these values into the expression:
\[ \frac{\sin 30^\circ - \cos 30^\circ + 1}{\sin 30^\circ + \cos 30^\circ - 1} = \frac{\frac{1}{2} - \frac{\sqrt{3}}{2} + 1}{\frac{1}{2} + \frac{\sqrt{3}}{2} - 1} \]

### Step 3: Simplify the numerator
Calculating the numerator:
\[ \frac{1}{2} - \frac{\sqrt{3}}{2} + 1 = \frac{1}{2} - \frac{\sqrt{3}}{2} + \frac{2}{2} = \frac{3 - \sqrt{3}}{2} \]

### Step 4: Simplify the denominator
Calculating the denominator:
\[ \frac{1}{2} + \frac{\sqrt{3}}{2} - 1 = \frac{1}{2} + \frac{\sqrt{3}}{2} - \frac{2}{2} = \frac{-1 + \sqrt{3}}{2} \]

### Step 5: Combine the results
Now we can combine the results:
\[ \frac{\frac{3 - \sqrt{3}}{2}}{\frac{-1 + \sqrt{3}}{2}} = \frac{3 - \sqrt{3}}{-1 + \sqrt{3}} \]

### Step 6: Rationalize the denominator
To simplify further, we can multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(3 - \sqrt{3})(\sqrt{3} + 1)}{(-1 + \sqrt{3})(\sqrt{3} + 1)} \]

Calculating the denominator:
\[ (-1 + \sqrt{3})(\sqrt{3} + 1) = -\sqrt{3} - 1 + 3 + \sqrt{3} = 2 \]

Calculating the numerator:
\[ (3 - \sqrt{3})(\sqrt{3} + 1) = 3\sqrt{3} + 3 - \sqrt{3} - 3 = 2\sqrt{3} \]

### Step 7: Final expression
Thus, we have:
\[ \frac{2\sqrt{3}}{2} = \sqrt{3} \]

### Conclusion
The value of the expression \(\frac{\sin \theta - \cos \theta + 1}{\sin \theta + \cos \theta - 1}\) when \(\theta = 30^\circ\) is \(\sqrt{3}\).