If `2 cos theta =sqrt(3) , cos theta xx tan theta = `?
A. 1
B. `sqrt(3)//3`
C. `sqrt(3)//2`
D. `1//2`

Options

A. A
B. D
C. B
D. C

Answer

To solve the problem, we start with the equation given:

1. **Given Equation**:
\[ 2 \cos \theta = \sqrt{3} \]

2. **Finding \(\cos \theta\)**:
To isolate \(\cos \theta\), we divide both sides of the equation by 2:
\[ \cos \theta = \frac{\sqrt{3}}{2} \]

3. **Identifying \(\theta\)**:
We recognize that \(\cos \theta = \frac{\sqrt{3}}{2}\) corresponds to \(\theta = 30^\circ\) (or \(\frac{\pi}{6}\) radians).

4. **Finding \(\tan \theta\)**:
We know that:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
For \(\theta = 30^\circ\):
- \(\sin 30^\circ = \frac{1}{2}\)
- \(\cos 30^\circ = \frac{\sqrt{3}}{2}\)

Therefore,
\[ \tan 30^\circ = \frac{\sin 30^\circ}{\cos 30^\circ} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \]

5. **Calculating \(\cos \theta \cdot \tan \theta\)**:
Now we can calculate \(\cos \theta \cdot \tan \theta\):
\[ \cos \theta \cdot \tan \theta = \cos \theta \cdot \frac{\sin \theta}{\cos \theta} = \sin \theta \]
Since we already know \(\sin 30^\circ = \frac{1}{2}\), we have:
\[ \cos \theta \cdot \tan \theta = \frac{1}{2} \]

6. **Final Answer**:
Thus, the value of \(\cos \theta \cdot \tan \theta\) is:
\[ \frac{1}{2} \]

The correct option is **D. \(\frac{1}{2}\)**.