`(1+ sin A- cos A)/( 1+ sin A + cos A)=`

Options

A. `sin ""A/2`
B. `cos ""A/2`
C. `tan ""A/2`
D. `cot ""A/2`

Answer

To solve the expression \(\frac{1 + \sin A - \cos A}{1 + \sin A + \cos A}\), we can simplify it step by step.

### Step 1: Rewrite the Expression
We start with the given expression:
\[ \frac{1 + \sin A - \cos A}{1 + \sin A + \cos A} \]

### Step 2: Use Trigonometric Identities
We can use the identities for \(1 - \cos A\) and \(1 + \cos A\):
- \(1 - \cos A = 2 \sin^2\left(\frac{A}{2}\right)\)
- \(1 + \cos A = 2 \cos^2\left(\frac{A}{2}\right)\)

### Step 3: Rewrite the Numerator and Denominator
Using the identities:
- The numerator becomes:
\[ 1 + \sin A - \cos A = 1 + 2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) - 2 \cos^2\left(\frac{A}{2}\right) \]
- The denominator becomes:
\[ 1 + \sin A + \cos A = 1 + 2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) + 2 \cos^2\left(\frac{A}{2}\right) \]

### Step 4: Simplify the Expression
Now, we can simplify both the numerator and the denominator:
- The numerator simplifies to:
\[ 1 + 2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) - 2 \cos^2\left(\frac{A}{2}\right) = (1 - 2 \cos^2\left(\frac{A}{2}\right)) + 2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) \]
- The denominator simplifies to:
\[ 1 + 2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) + 2 \cos^2\left(\frac{A}{2}\right) = (1 + 2 \cos^2\left(\frac{A}{2}\right)) + 2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) \]

### Step 5: Factor Out Common Terms
Now we can factor out common terms from the numerator and denominator:
\[ \frac{(1 - 2 \cos^2\left(\frac{A}{2}\right)) + 2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right)}{(1 + 2 \cos^2\left(\frac{A}{2}\right)) + 2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right)} \]

### Step 6: Final Simplification
After canceling out common terms, we can simplify the expression to:
\[ \frac{\sin\left(\frac{A}{2}\right)}{\cos\left(\frac{A}{2}\right)} = \tan\left(\frac{A}{2}\right) \]

### Conclusion
Thus, the final simplified expression is:
\[ \frac{1 + \sin A - \cos A}{1 + \sin A + \cos A} = \tan\left(\frac{A}{2}\right) \]