If x tan ` 30^(@) = (sin 30^(@) + cos 60^(@))/( tan 60^(@) + sin 60^(@))` , then the value of x is :

Options

A. `(2)/(3)`
B. `(2)/(sqrt(3))`
C. `(2)/(3 sqrt(3))`
D. `(3)/(2)`

Answer

To solve the equation \( x \tan 30^\circ = \frac{\sin 30^\circ + \cos 60^\circ}{\tan 60^\circ + \sin 60^\circ} \), we will follow these steps:

### Step 1: Substitute the trigonometric values
We know the following values:
- \( \tan 30^\circ = \frac{1}{\sqrt{3}} \)
- \( \sin 30^\circ = \frac{1}{2} \)
- \( \cos 60^\circ = \frac{1}{2} \)
- \( \tan 60^\circ = \sqrt{3} \)
- \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)

Substituting these values into the equation gives us:
\[ x \cdot \frac{1}{\sqrt{3}} = \frac{\frac{1}{2} + \frac{1}{2}}{\sqrt{3} + \frac{\sqrt{3}}{2}} \]

### Step 2: Simplify the left-hand side
The left-hand side becomes:
\[ \frac{x}{\sqrt{3}} \]

### Step 3: Simplify the right-hand side
Now, simplify the right-hand side:
\[ \frac{\frac{1}{2} + \frac{1}{2}}{\sqrt{3} + \frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3} + \frac{\sqrt{3}}{2}} \]

To simplify the denominator, we can find a common denominator:
\[ \sqrt{3} + \frac{\sqrt{3}}{2} = \frac{2\sqrt{3}}{2} + \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2} \]

Thus, the right-hand side becomes:
\[ \frac{1}{\frac{3\sqrt{3}}{2}} = \frac{2}{3\sqrt{3}} \]

### Step 4: Set the two sides equal
Now we have:
\[ \frac{x}{\sqrt{3}} = \frac{2}{3\sqrt{3}} \]

### Step 5: Solve for \( x \)
To solve for \( x \), we can cross-multiply:
\[ x \cdot 3\sqrt{3} = 2\sqrt{3} \]
Dividing both sides by \( \sqrt{3} \):
\[ 3x = 2 \]
Now, divide by 3:
\[ x = \frac{2}{3} \]

### Final Answer
The value of \( x \) is \( \frac{2}{3} \).

---