Evaluate : `(sin^(2)30^(@)+sin^(2)45^(@)-4cot^(2)60^(@))/(2sin30^(@)cos30^(@)+(1)/(2)tan60^(@))`

Answer

To evaluate the expression

\[ \frac{\sin^2 30^\circ + \sin^2 45^\circ - 4 \cot^2 60^\circ}{2 \sin 30^\circ \cos 30^\circ + \frac{1}{2} \tan 60^\circ} \]

we will follow these steps:

### Step 1: Calculate \(\sin^2 30^\circ\)
\[ \sin 30^\circ = \frac{1}{2} \implies \sin^2 30^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 2: Calculate \(\sin^2 45^\circ\)
\[ \sin 45^\circ = \frac{1}{\sqrt{2}} \implies \sin^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

### Step 3: Calculate \(\cot^2 60^\circ\)
\[ \cot 60^\circ = \frac{1}{\tan 60^\circ} = \frac{1}{\sqrt{3}} \implies \cot^2 60^\circ = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3} \]
Thus,
\[ 4 \cot^2 60^\circ = 4 \times \frac{1}{3} = \frac{4}{3} \]

### Step 4: Substitute values into the numerator
Now substituting the values we calculated into the numerator:
\[ \text{Numerator} = \frac{1}{4} + \frac{1}{2} - \frac{4}{3} \]

To combine these fractions, we need a common denominator. The least common multiple of 4, 2, and 3 is 12.
\[ \frac{1}{4} = \frac{3}{12}, \quad \frac{1}{2} = \frac{6}{12}, \quad \frac{4}{3} = \frac{16}{12} \]
Thus,
\[ \text{Numerator} = \frac{3}{12} + \frac{6}{12} - \frac{16}{12} = \frac{9 - 16}{12} = \frac{-7}{12} \]

### Step 5: Calculate \(2 \sin 30^\circ \cos 30^\circ\)
\[ \sin 30^\circ = \frac{1}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2} \]
Thus,
\[ 2 \sin 30^\circ \cos 30^\circ = 2 \times \frac{1}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \]

### Step 6: Calculate \(\tan 60^\circ\)
\[ \tan 60^\circ = \sqrt{3} \]
Thus,
\[ \frac{1}{2} \tan 60^\circ = \frac{1}{2} \times \sqrt{3} = \frac{\sqrt{3}}{2} \]

### Step 7: Substitute values into the denominator
Now substituting the values we calculated into the denominator:
\[ \text{Denominator} = \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} = \sqrt{3} \]

### Step 8: Final calculation
Now we can substitute the numerator and denominator into the original expression:
\[ \frac{\text{Numerator}}{\text{Denominator}} = \frac{\frac{-7}{12}}{\sqrt{3}} = \frac{-7}{12\sqrt{3}} \]

### Final Answer
Thus, the final answer is:
\[ \frac{-7}{12\sqrt{3}} \]