`(sin30^(@)cos45^(@))/(tan 60^(@))` is equal to

Options

A. `(2)/(sqrt(3))`
B. `(sqrt(6))/(12)`
C. `2sqrt(3)`
D. `sqrt((2)/(3))`

Answer

To solve the expression \((\sin 30^\circ \cdot \cos 45^\circ) / \tan 60^\circ\), we will follow these steps:

### Step 1: Write down the values of the trigonometric ratios.
- \(\sin 30^\circ = \frac{1}{2}\)
- \(\cos 45^\circ = \frac{1}{\sqrt{2}}\)
- \(\tan 60^\circ = \sqrt{3}\)

### Step 2: Substitute the values into the expression.
Now we substitute these values into the expression:

\[ \frac{\sin 30^\circ \cdot \cos 45^\circ}{\tan 60^\circ} = \frac{\left(\frac{1}{2}\right) \cdot \left(\frac{1}{\sqrt{2}}\right)}{\sqrt{3}} \]

### Step 3: Simplify the numerator.
Multiply the values in the numerator:

\[ \frac{1}{2} \cdot \frac{1}{\sqrt{2}} = \frac{1}{2\sqrt{2}} \]

### Step 4: Rewrite the expression.
Now the expression can be rewritten as:

\[ \frac{\frac{1}{2\sqrt{2}}}{\sqrt{3}} = \frac{1}{2\sqrt{2} \cdot \sqrt{3}} \]

### Step 5: Simplify further.
We can simplify the denominator:

\[ 2\sqrt{2} \cdot \sqrt{3} = 2\sqrt{6} \]

So the expression now becomes:

\[ \frac{1}{2\sqrt{6}} \]

### Step 6: Rationalize the denominator.
To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{6}\):

\[ \frac{1 \cdot \sqrt{6}}{2\sqrt{6} \cdot \sqrt{6}} = \frac{\sqrt{6}}{12} \]

### Final Answer:
Thus, the final answer is:

\[ \frac{\sqrt{6}}{12} \]

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