`((2sinA)(1+sinA))/(1+sinA+cosA)` is equal to:

Options

A. 1 + sin A cos A
B. 1 + sin A - cos A
C. 1 - sin A cos A
D. 1 + cos A - sin A

Answer

To solve the expression \(\frac{(2\sin A)(1+\sin A)}{1+\sin A+\cos A}\), we will simplify it step by step.

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{(2\sin A)(1+\sin A)}{1+\sin A+\cos A} \]

### Step 2: Expand the numerator
Next, we expand the numerator:
\[ 2\sin A(1+\sin A) = 2\sin A + 2\sin^2 A \]
So now our expression looks like:
\[ \frac{2\sin A + 2\sin^2 A}{1+\sin A+\cos A} \]

### Step 3: Factor out the numerator
We can factor out \(2\) from the numerator:
\[ = \frac{2(\sin A + \sin^2 A)}{1+\sin A+\cos A} \]

### Step 4: Simplify the expression
Now, we can simplify the expression further. Notice that \(\sin^2 A\) can be rewritten using the Pythagorean identity:
\[ \sin^2 A = 1 - \cos^2 A \]
However, in this case, we will keep it as is for simplicity.

### Step 5: Analyze the denominator
The denominator \(1 + \sin A + \cos A\) does not simplify directly with the numerator, so we will leave it as is.

### Step 6: Final expression
Thus, the expression simplifies to:
\[ \frac{2(\sin A + \sin^2 A)}{1+\sin A+\cos A} \]

### Conclusion
The final simplified form of the expression \(\frac{(2\sin A)(1+\sin A)}{1+\sin A+\cos A}\) is:
\[ \frac{2(\sin A + \sin^2 A)}{1+\sin A+\cos A} \]