The is the value of `(sin^2 30^@)/(cos^2 30^@)+(cos^2 30^@)/(sin^2 30^@)` ?

Answer

To solve the expression \(\frac{\sin^2 30^\circ}{\cos^2 30^\circ} + \frac{\cos^2 30^\circ}{\sin^2 30^\circ}\), we will follow these steps:

### Step 1: Find the values of \(\sin 30^\circ\) and \(\cos 30^\circ\)
We know that:
\[ \sin 30^\circ = \frac{1}{2} \]
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]

### Step 2: Substitute these values into the expression
Now we substitute these values into the expression:
\[ \frac{\sin^2 30^\circ}{\cos^2 30^\circ} + \frac{\cos^2 30^\circ}{\sin^2 30^\circ} = \frac{\left(\frac{1}{2}\right)^2}{\left(\frac{\sqrt{3}}{2}\right)^2} + \frac{\left(\frac{\sqrt{3}}{2}\right)^2}{\left(\frac{1}{2}\right)^2} \]

### Step 3: Simplify the squares
Calculating the squares:
\[ \left(\frac{1}{2}\right)^2 = \frac{1}{4}, \quad \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]
Now substituting these values back into the expression:
\[ \frac{\frac{1}{4}}{\frac{3}{4}} + \frac{\frac{3}{4}}{\frac{1}{4}} \]

### Step 4: Simplify the fractions
Now we simplify each term:
\[ \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{4} \cdot \frac{4}{3} = \frac{1}{3} \]
\[ \frac{\frac{3}{4}}{\frac{1}{4}} = \frac{3}{4} \cdot \frac{4}{1} = 3 \]

### Step 5: Combine the results
Now we combine the results:
\[ \frac{1}{3} + 3 = \frac{1}{3} + \frac{9}{3} = \frac{10}{3} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{10}{3} \]