Let `x =(log_(sin 30^(@))cos 30^(@))(log_(cos30^(@))cot30^(@)) (log_(cot 30^(@))coses 30^(@))` and `y = 3^(log_(cot 30^(@))(|x|+3))` then value of `y` is equal to-

Options

A. 4
B. 16
C. 12
D. 10

Answer

To solve the given problem, we will calculate the value of \( x \) and then use it to find \( y \).

### Step 1: Calculate \( x \)

The expression for \( x \) is given as:
\[ x = \log_{\sin 30^\circ} \cos 30^\circ \cdot \log_{\cos 30^\circ} \cot 30^\circ \cdot \log_{\cot 30^\circ} \csc 30^\circ \]

Using the known values:
- \( \sin 30^\circ = \frac{1}{2} \)
- \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)
- \( \cot 30^\circ = \frac{1}{\tan 30^\circ} = \sqrt{3} \)
- \( \csc 30^\circ = \frac{1}{\sin 30^\circ} = 2 \)

We can rewrite \( x \) using the change of base formula:
\[ \log_a b = \frac{\log_c b}{\log_c a} \]

Thus, we can express each logarithm in terms of natural logarithm (or any common base):
1. \( \log_{\sin 30^\circ} \cos 30^\circ = \frac{\log \cos 30^\circ}{\log \sin 30^\circ} \)
2. \( \log_{\cos 30^\circ} \cot 30^\circ = \frac{\log \cot 30^\circ}{\log \cos 30^\circ} \)
3. \( \log_{\cot 30^\circ} \csc 30^\circ = \frac{\log \csc 30^\circ}{\log \cot 30^\circ} \)

Substituting these into \( x \):
\[ x = \frac{\log \cos 30^\circ}{\log \sin 30^\circ} \cdot \frac{\log \cot 30^\circ}{\log \cos 30^\circ} \cdot \frac{\log \csc 30^\circ}{\log \cot 30^\circ} \]

### Step 2: Simplify \( x \)

Notice that the terms in the product will cancel out:
\[ x = \frac{\log \csc 30^\circ}{\log \sin 30^\circ} \]
Now, substituting the known values:
\[ \csc 30^\circ = 2 \quad \text{and} \quad \sin 30^\circ = \frac{1}{2} \]
Thus,
\[ x = \frac{\log 2}{\log \left(\frac{1}{2}\right)} = \frac{\log 2}{-\log 2} = -1 \]

### Step 3: Calculate \( y \)

Now substituting \( x \) into the expression for \( y \):
\[ y = 3^{\log_{\cot 30^\circ} (|x| + 3)} \]
Since \( |x| = 1 \):
\[ y = 3^{\log_{\cot 30^\circ} (1 + 3)} = 3^{\log_{\cot 30^\circ} 4} \]
Using the change of base formula again:
\[ y = 3^{\frac{\log 4}{\log \cot 30^\circ}} \]
Since \( \cot 30^\circ = \sqrt{3} \):
\[ y = 3^{\frac{\log 4}{\log \sqrt{3}}} = 3^{\frac{\log 4}{\frac{1}{2} \log 3}} = 3^{\frac{2 \log 4}{\log 3}} = 4^{\frac{2}{\log 3}} = 4^{\log_3 4} = 3^2 = 9 \]

### Final Answer:
Thus, the value of \( y \) is:
\[ \boxed{9} \]