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 given problem step by step, we start with the equation:

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

### Step 1: Cross-multiply to eliminate the fraction

Cross-multiplying gives us:

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

This simplifies to:

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

### Step 2: Rearrange the equation

Now, we can rearrange the equation to isolate terms involving \(\sin \alpha\) and \(\cos \alpha\):

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

This simplifies to:

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

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

Now, we can isolate \(\cos \alpha\):

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

Dividing through by 3 gives:

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

### Step 4: Substitute into the expression we need to find

We need to find the value of:

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

Substituting \(\cos \alpha\) into this expression:

\[ \frac{1 + \left(\frac{5 \sin \alpha - 3}{3}\right) + \sin \alpha}{1 + \sin \alpha} \]

### Step 5: Simplify the numerator

The numerator becomes:

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

### Step 6: Final expression

Now we can express our fraction:

\[ \frac{\frac{8 \sin \alpha}{3}}{1 + \sin \alpha} \]

This simplifies to:

\[ \frac{8 \sin \alpha}{3(1 + \sin \alpha)} \]

### Step 7: Substitute \(\sin \alpha\) back using the original equation

From the earlier steps, we know that:

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

Substituting this into our expression gives us the final value.

### Conclusion

After substituting and simplifying, we find that:

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