`(tanA-secA+1)/(tanA+secA-1)=?`

Options

A. `(1+sinA)/(cosA)`
B. `(1-sinA)/(cosA)`
C. `(1+cosA)/(sinA)`
D. `(1-cosA)/(sinA)`

Answer

To solve the expression \((\tan A - \sec A + 1)/(\tan A + \sec A - 1)\), we will follow these steps:

### Step 1: Rewrite \(\tan A\) and \(\sec A\)
Recall the definitions of \(\tan A\) and \(\sec A\):
\[ \tan A = \frac{\sin A}{\cos A}, \quad \sec A = \frac{1}{\cos A} \]
Substituting these into the expression gives:
\[ \frac{\frac{\sin A}{\cos A} - \frac{1}{\cos A} + 1}{\frac{\sin A}{\cos A} + \frac{1}{\cos A} - 1} \]

### Step 2: Combine the terms in the numerator
The numerator becomes:
\[ \frac{\sin A - 1 + \cos A}{\cos A} \]
This simplifies to:
\[ \frac{\sin A + \cos A - 1}{\cos A} \]

### Step 3: Combine the terms in the denominator
The denominator becomes:
\[ \frac{\sin A + 1 - \cos A}{\cos A} \]
This simplifies to:
\[ \frac{\sin A + 1 - \cos A}{\cos A} \]

### Step 4: Rewrite the entire expression
Now we can rewrite the entire expression:
\[ \frac{\frac{\sin A + \cos A - 1}{\cos A}}{\frac{\sin A + 1 - \cos A}{\cos A}} = \frac{\sin A + \cos A - 1}{\sin A + 1 - \cos A} \]

### Step 5: Simplify the expression
We can simplify this further by factoring or manipulating the numerator and denominator. Let's rewrite it:
\[ \frac{\sin A + \cos A - 1}{\sin A + 1 - \cos A} \]

### Step 6: Use the identity
We can use the Pythagorean identity \(\sin^2 A + \cos^2 A = 1\) to express the terms differently, but we will also notice that the expression can be simplified directly by substituting values or recognizing patterns.

### Final Result
After simplification, we find that:
\[ \frac{\sin A + \cos A - 1}{\sin A + 1 - \cos A} = \sec A - \tan A \]

Thus, the final simplified result of the expression \((\tan A - \sec A + 1)/(\tan A + \sec A - 1)\) is:
\[ \sec A - \tan A \]