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

Answer

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

### Step 1: Evaluate \( \tan^2 45^\circ \)
We know that:
\[ \tan 45^\circ = 1 \]
Thus,
\[ \tan^2 45^\circ = 1^2 = 1 \]

### Step 2: Calculate \( 2 \tan^2 45^\circ \)
Now, substituting the value from Step 1:
\[ 2 \tan^2 45^\circ = 2 \times 1 = 2 \]

### Step 3: Evaluate \( \cos^2 30^\circ \)
We know that:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]
Thus,
\[ \cos^2 30^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

### Step 4: Evaluate \( \sin^2 60^\circ \)
We know that:
\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \]
Thus,
\[ \sin^2 60^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

### Step 5: Substitute values into the expression
Now we substitute the values we calculated into the original expression:
\[ 2 \tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ = 2 + \frac{3}{4} - \frac{3}{4} \]

### Step 6: Simplify the expression
Since \( \frac{3}{4} - \frac{3}{4} = 0 \), we have:
\[ 2 + 0 = 2 \]

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