If `sinx +cosx=(sqrt7)/(2)`, where `x in [0, (pi)/(4)],` then the value of `tan.(x)/(2)` is equal to

Options

A. `(3-sqrt7)/(3)`
B. `(sqrt7-2)/(3)`
C. `(4-sqrt7)/(4)`
D. `(5-sqrt3)/(2)`

Answer

To solve the problem where \( \sin x + \cos x = \frac{\sqrt{7}}{2} \) and \( x \in [0, \frac{\pi}{4}] \), we need to find the value of \( \tan\left(\frac{x}{2}\right) \).

### Step 1: Rewrite the equation
We start with the equation:
\[ \sin x + \cos x = \frac{\sqrt{7}}{2} \]

### Step 2: Use the identity for \(\sin x + \cos x\)
We know that:
\[ \sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) \]
Thus, we can rewrite the equation as:
\[ \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) = \frac{\sqrt{7}}{2} \]

### Step 3: Solve for \(\sin\left(x + \frac{\pi}{4}\right)\)
Dividing both sides by \(\sqrt{2}\):
\[ \sin\left(x + \frac{\pi}{4}\right) = \frac{\sqrt{7}}{2\sqrt{2}} = \frac{\sqrt{14}}{4} \]

### Step 4: Find the angle
The angle whose sine is \(\frac{\sqrt{14}}{4}\) can be found using the inverse sine function:
\[ x + \frac{\pi}{4} = \arcsin\left(\frac{\sqrt{14}}{4}\right) \]
Thus,
\[ x = \arcsin\left(\frac{\sqrt{14}}{4}\right) - \frac{\pi}{4} \]

### Step 5: Use the half-angle formula for tangent
We can express \(\tan\left(\frac{x}{2}\right)\) using the half-angle formula:
\[ \tan\left(\frac{x}{2}\right) = \frac{1 - \cos x}{\sin x} \]

### Step 6: Find \(\sin x\) and \(\cos x\)
From the original equation, we can square both sides:
\[ (\sin x + \cos x)^2 = \left(\frac{\sqrt{7}}{2}\right)^2 \]
Expanding the left side:
\[ \sin^2 x + 2\sin x \cos x + \cos^2 x = \frac{7}{4} \]
Using \(\sin^2 x + \cos^2 x = 1\):
\[ 1 + 2\sin x \cos x = \frac{7}{4} \]
Thus,
\[ 2\sin x \cos x = \frac{7}{4} - 1 = \frac{3}{4} \]
This gives:
\[ \sin x \cos x = \frac{3}{8} \]

### Step 7: Find \(\tan\left(\frac{x}{2}\right)\)
Using the identity \( \sin x = 2 \sin\left(\frac{x}{2}\right) \cos\left(\frac{x}{2}\right) \) and \( \cos x = \cos^2\left(\frac{x}{2}\right) - \sin^2\left(\frac{x}{2}\right) \):
\[ \tan\left(\frac{x}{2}\right) = \frac{\sin x}{1 + \cos x} \]
Substituting the values we derived, we can compute \( \tan\left(\frac{x}{2}\right) \).

### Final Step: Calculate the value
After substituting the values and simplifying, we find:
\[ \tan\left(\frac{x}{2}\right) = \frac{\sqrt{7} - 2}{3} \]

Thus, the final answer is:
\[ \boxed{\frac{\sqrt{7} - 2}{3}} \]