The expression `(cotx+cosecx-1)/(cotx-cosecx+1)` is equal to:

Options

A. A) `(sinx)/(1-cosx)`
B. B) `(1-cosx)/(sinx)`
C. C) `(1+cosx)/(sinx)`
D. D) `(sinx)/(1+cosx)`

Answer

To solve the expression \((\cot x + \csc x - 1)/(\cot x - \csc x + 1)\), we will follow these steps:

### Step 1: Rewrite cotangent and cosecant in terms of sine and cosine
Recall the definitions:
- \(\cot x = \frac{\cos x}{\sin x}\)
- \(\csc x = \frac{1}{\sin x}\)

Substituting these into the expression gives:
\[ \frac{\left(\frac{\cos x}{\sin x} + \frac{1}{\sin x} - 1\right)}{\left(\frac{\cos x}{\sin x} - \frac{1}{\sin x} + 1\right)} \]

### Step 2: Combine the terms in the numerator and the denominator
In the numerator:
\[ \frac{\cos x + 1 - \sin x}{\sin x} \]
In the denominator:
\[ \frac{\cos x - 1 + \sin x}{\sin x} \]

So the expression simplifies to:
\[ \frac{\cos x + 1 - \sin x}{\cos x - 1 + \sin x} \]

### Step 3: Simplify the expression
Now, we can multiply both the numerator and the denominator by \(\sin x\) to eliminate the fraction:
\[ \frac{\cos x + 1 - \sin x}{\cos x - 1 + \sin x} \]

### Step 4: Factor the numerator and denominator
Notice that we can rearrange the terms:
- Numerator: \((\cos x - \sin x) + 1\)
- Denominator: \((\sin x - \cos x) + 1\)

This gives us:
\[ \frac{(\cos x - \sin x) + 1}{(\sin x - \cos x) + 1} \]

### Step 5: Recognize the structure
We can see that the numerator and denominator have a similar structure. We can rewrite the expression as:
\[ \frac{1 + \cos x - \sin x}{1 - \cos x + \sin x} \]

### Step 6: Final simplification
This expression can be further simplified, but we can also check for specific values of \(x\) to find a numerical result. For example, if we let \(x = 0\):
- \(\cot(0) = \infty\) and \(\csc(0) = \infty\), which leads to an indeterminate form.

However, if we evaluate at \(x = \frac{\pi}{4}\):
- \(\cot\left(\frac{\pi}{4}\right) = 1\) and \(\csc\left(\frac{\pi}{4}\right) = \sqrt{2}\)

Substituting these values gives:
\[ \frac{1 + \sqrt{2} - 1}{1 - \sqrt{2} + 1} = \frac{\sqrt{2}}{2 - \sqrt{2}} \]

### Conclusion
The expression simplifies to:
\[ \frac{1 + \cos x - \sin x}{1 - \cos x + \sin x} \]