Evaluate : `2sin^(2)30^(@)tan60^(@)-3cos^(2)30^(@)sec60^(@)`

Answer

To evaluate the expression \(2\sin^2 30^\circ \tan 60^\circ - 3\cos^2 30^\circ \sec 60^\circ\), we will follow these steps:

### Step 1: Find the trigonometric values
We need to know the values of the trigonometric functions involved:
- \(\sin 30^\circ = \frac{1}{2}\)
- \(\cos 30^\circ = \frac{\sqrt{3}}{2}\)
- \(\tan 60^\circ = \sqrt{3}\)
- \(\sec 60^\circ = 2\)

### Step 2: Substitute the values into the expression
Now we will substitute these values into the expression:
\[ 2\sin^2 30^\circ \tan 60^\circ - 3\cos^2 30^\circ \sec 60^\circ \]
Substituting the values:
\[ = 2\left(\frac{1}{2}\right)^2 \cdot \sqrt{3} - 3\left(\frac{\sqrt{3}}{2}\right)^2 \cdot 2 \]

### Step 3: Calculate \(\sin^2 30^\circ\) and \(\cos^2 30^\circ\)
Calculating \(\sin^2 30^\circ\) and \(\cos^2 30^\circ\):
\[ \sin^2 30^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
\[ \cos^2 30^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

### Step 4: Substitute back into the expression
Now substituting these squared values back into the expression:
\[ = 2 \cdot \frac{1}{4} \cdot \sqrt{3} - 3 \cdot \frac{3}{4} \cdot 2 \]
\[ = \frac{2}{4}\sqrt{3} - \frac{9}{4} \]
\[ = \frac{1}{2}\sqrt{3} - \frac{9}{4} \]

### Step 5: Find a common denominator
To combine the terms, we need a common denominator:
The common denominator between 2 and 4 is 4. Therefore, we rewrite \(\frac{1}{2}\sqrt{3}\) as:
\[ \frac{1}{2}\sqrt{3} = \frac{2\sqrt{3}}{4} \]
Now substituting this back:
\[ = \frac{2\sqrt{3}}{4} - \frac{9}{4} \]

### Step 6: Combine the fractions
Now we can combine the fractions:
\[ = \frac{2\sqrt{3} - 9}{4} \]

### Final Answer
Thus, the final answer is:
\[ \frac{2\sqrt{3} - 9}{4} \]