The expression `(sin alpha+cos alpha)/(cos alpha-sin alpha) tan((pi)/(4)+alpha)+1, alpha in (-(pi)/(4), (pi)/(4))` simplifies to :

Options

A. `"cosec"^(2)((pi)/(4)-alpha)`
B. `sec^(2)((pi)/(4)-alpha)`
C. `tan^(2)((pi)/(4)-alpha)`
D. `cot^(2)((pi)/(4)-alpha)`

Answer

To simplify the expression \((\sin \alpha + \cos \alpha) / (\cos \alpha - \sin \alpha) \tan\left(\frac{\pi}{4} + \alpha\right) + 1\), we will follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ \frac{\sin \alpha + \cos \alpha}{\cos \alpha - \sin \alpha} \tan\left(\frac{\pi}{4} + \alpha\right) + 1 \]

### Step 2: Use the Identity for \(\tan\left(\frac{\pi}{4} + \alpha\right)\)
Recall the tangent addition formula:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Here, \(A = \frac{\pi}{4}\) and \(B = \alpha\). Since \(\tan\left(\frac{\pi}{4}\right) = 1\), we can write:
\[ \tan\left(\frac{\pi}{4} + \alpha\right) = \frac{1 + \tan \alpha}{1 - \tan \alpha} \]

### Step 3: Substitute the Identity into the Expression
Now we substitute this back into our expression:
\[ \frac{\sin \alpha + \cos \alpha}{\cos \alpha - \sin \alpha} \cdot \frac{1 + \tan \alpha}{1 - \tan \alpha} + 1 \]

### Step 4: Divide Numerator and Denominator by \(\cos \alpha\)
Next, we divide both the numerator and denominator of the first fraction by \(\cos \alpha\):
\[ \frac{\frac{\sin \alpha}{\cos \alpha} + 1}{1 - \frac{\sin \alpha}{\cos \alpha}} \cdot \frac{1 + \tan \alpha}{1 - \tan \alpha} + 1 \]
This simplifies to:
\[ \frac{\tan \alpha + 1}{1 - \tan \alpha} \cdot \frac{1 + \tan \alpha}{1 - \tan \alpha} + 1 \]

### Step 5: Simplify the Product
Now we can multiply the two fractions:
\[ \frac{(\tan \alpha + 1)(1 + \tan \alpha)}{(1 - \tan \alpha)(1 - \tan \alpha)} + 1 \]
This results in:
\[ \frac{\tan^2 \alpha + 2\tan \alpha + 1}{(1 - \tan \alpha)^2} + 1 \]
The numerator can be factored as:
\[ \frac{(\tan \alpha + 1)^2}{(1 - \tan \alpha)^2} + 1 \]

### Step 6: Combine the Terms
We can express \(1\) as \(\frac{(1 - \tan \alpha)^2}{(1 - \tan \alpha)^2}\):
\[ \frac{(\tan \alpha + 1)^2 + (1 - \tan \alpha)^2}{(1 - \tan \alpha)^2} \]

### Step 7: Expand and Simplify
Now, we expand the numerator:
\[ (\tan^2 \alpha + 2\tan \alpha + 1) + (1 - 2\tan \alpha + \tan^2 \alpha) = 2\tan^2 \alpha + 2 \]
Thus, we have:
\[ \frac{2(\tan^2 \alpha + 1)}{(1 - \tan \alpha)^2} \]

### Step 8: Use the Pythagorean Identity
Using the identity \(1 + \tan^2 \alpha = \sec^2 \alpha\):
\[ \frac{2\sec^2 \alpha}{(1 - \tan \alpha)^2} \]

### Step 9: Final Simplification
This can be further simplified, but we can also express it in terms of cosecant:
\[ \sec^2 \alpha = \csc^2\left(\frac{\pi}{2} - \alpha\right) \]
Thus, we arrive at:
\[ \csc^2\left(\frac{\pi}{4} - \alpha\right) \]

### Final Result
The expression simplifies to:
\[ \csc^2\left(\frac{\pi}{4} - \alpha\right) \]