Evaluate : `sin^2 60^@+ 2tan 45^@-cos^2 30^@`

Answer

To evaluate the expression \( \sin^2 60^\circ + 2\tan 45^\circ - \cos^2 30^\circ \), we will follow these steps:

### Step 1: Find \( \sin^2 60^\circ \)
The value of \( \sin 60^\circ \) is \( \frac{\sqrt{3}}{2} \). Therefore,
\[ \sin^2 60^\circ = \left( \frac{\sqrt{3}}{2} \right)^2 = \frac{3}{4} \]

### Step 2: Find \( \tan 45^\circ \)
The value of \( \tan 45^\circ \) is \( 1 \). Therefore,
\[ 2\tan 45^\circ = 2 \times 1 = 2 \]

### Step 3: Find \( \cos^2 30^\circ \)
The value of \( \cos 30^\circ \) is \( \frac{\sqrt{3}}{2} \). Therefore,
\[ \cos^2 30^\circ = \left( \frac{\sqrt{3}}{2} \right)^2 = \frac{3}{4} \]

### Step 4: Substitute the values into the expression
Now we substitute the values we found into the original expression:
\[ \sin^2 60^\circ + 2\tan 45^\circ - \cos^2 30^\circ = \frac{3}{4} + 2 - \frac{3}{4} \]

### Step 5: Simplify the expression
Notice that \( \frac{3}{4} \) and \( -\frac{3}{4} \) will cancel each other out:
\[ \frac{3}{4} - \frac{3}{4} + 2 = 0 + 2 = 2 \]

### Final Answer
Thus, the value of the expression \( \sin^2 60^\circ + 2\tan 45^\circ - \cos^2 30^\circ \) is:
\[ \boxed{2} \]