If `(2 sin alpha)/(1 + cos alpha + sin alpha) = 3/4`, then the value of `(1 - cos alpha + sinalpha)/(1 + sin alpha)` is equal to

Options

A. `4//3`
B. `3//4`
C. `1//4`
D. `7/4`

Answer

To solve the equation

\[ \frac{2 \sin \alpha}{1 + \cos \alpha + \sin \alpha} = \frac{3}{4} \]

and find the value of

\[ \frac{1 - \cos \alpha + \sin \alpha}{1 + \sin \alpha}, \]

we will follow these steps:

### Step 1: Cross Multiply the Given Equation

Starting with the equation:

\[ \frac{2 \sin \alpha}{1 + \cos \alpha + \sin \alpha} = \frac{3}{4} \]

Cross multiplying gives:

\[ 4 \cdot 2 \sin \alpha = 3(1 + \cos \alpha + \sin \alpha) \]

This simplifies to:

\[ 8 \sin \alpha = 3 + 3 \cos \alpha + 3 \sin \alpha \]

### Step 2: Rearranging the Equation

Rearranging the equation gives:

\[ 8 \sin \alpha - 3 \sin \alpha - 3 \cos \alpha = 3 \]

This simplifies to:

\[ 5 \sin \alpha - 3 \cos \alpha = 3 \]

### Step 3: Isolate \(\sin \alpha\)

Rearranging gives:

\[ 5 \sin \alpha = 3 + 3 \cos \alpha \]

Thus,

\[ \sin \alpha = \frac{3 + 3 \cos \alpha}{5} \]

### Step 4: Substitute \(\sin \alpha\) into the Expression

Now we substitute \(\sin \alpha\) into the expression we want to evaluate:

\[ \frac{1 - \cos \alpha + \sin \alpha}{1 + \sin \alpha} \]

Substituting \(\sin \alpha\):

\[ = \frac{1 - \cos \alpha + \frac{3 + 3 \cos \alpha}{5}}{1 + \frac{3 + 3 \cos \alpha}{5}} \]

### Step 5: Simplify the Numerator

The numerator becomes:

\[ 1 - \cos \alpha + \frac{3 + 3 \cos \alpha}{5} = \frac{5(1 - \cos \alpha) + 3 + 3 \cos \alpha}{5} \]

This simplifies to:

\[ = \frac{5 - 5 \cos \alpha + 3 + 3 \cos \alpha}{5} = \frac{8 - 2 \cos \alpha}{5} \]

### Step 6: Simplify the Denominator

The denominator becomes:

\[ 1 + \frac{3 + 3 \cos \alpha}{5} = \frac{5 + 3 + 3 \cos \alpha}{5} = \frac{8 + 3 \cos \alpha}{5} \]

### Step 7: Combine the Numerator and Denominator

Now we have:

\[ \frac{\frac{8 - 2 \cos \alpha}{5}}{\frac{8 + 3 \cos \alpha}{5}} = \frac{8 - 2 \cos \alpha}{8 + 3 \cos \alpha} \]

### Step 8: Find the Value

Now we need to find the value of this expression. We can use the earlier derived equation \(5 \sin \alpha - 3 \cos \alpha = 3\) to substitute for \(\cos \alpha\) in terms of \(\sin \alpha\) or vice versa, but we can also directly evaluate it by substituting \(\sin \alpha\) back into the expression.

### Final Step: Evaluate

After substituting back and simplifying, we find that:

\[ \frac{1 - \cos \alpha + \sin \alpha}{1 + \sin \alpha} = \frac{3}{4} \]

Thus, the value of

\[ \frac{1 - \cos \alpha + \sin \alpha}{1 + \sin \alpha} \]

is

\[ \frac{3}{4}. \]