Evaluate: `(sin^(4) 60^(@) + sec^(4) 30^(@)) - 2 (cos^(2) 45^(@) - sin^(2) 90^(@))`

Answer

To evaluate the expression `(sin^4 60° + sec^4 30°) - 2 (cos^2 45° - sin^2 90°)`, we will follow these steps:

### Step 1: Calculate \( \sin 60° \) and \( \sec 30° \)
We know:
- \( \sin 60° = \frac{\sqrt{3}}{2} \)
- \( \sec 30° = \frac{1}{\cos 30°} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \)

### Step 2: Calculate \( \sin^4 60° \) and \( \sec^4 30° \)
Now we will calculate:
- \( \sin^4 60° = \left(\frac{\sqrt{3}}{2}\right)^4 = \frac{3^2}{2^4} = \frac{9}{16} \)
- \( \sec^4 30° = \left(\frac{2}{\sqrt{3}}\right)^4 = \frac{2^4}{(\sqrt{3})^4} = \frac{16}{9} \)

### Step 3: Substitute into the expression
Now substitute these values into the expression:
\[ \sin^4 60° + \sec^4 30° = \frac{9}{16} + \frac{16}{9} \]

### Step 4: Find a common denominator and add
The common denominator for \( \frac{9}{16} \) and \( \frac{16}{9} \) is \( 144 \):
- Convert \( \frac{9}{16} \) to \( \frac{81}{144} \) (by multiplying numerator and denominator by 9)
- Convert \( \frac{16}{9} \) to \( \frac{256}{144} \) (by multiplying numerator and denominator by 16)

Now add them:
\[ \frac{81}{144} + \frac{256}{144} = \frac{337}{144} \]

### Step 5: Calculate \( \cos^2 45° \) and \( \sin^2 90° \)
Next, we calculate:
- \( \cos 45° = \frac{1}{\sqrt{2}} \) so \( \cos^2 45° = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \)
- \( \sin 90° = 1 \) so \( \sin^2 90° = 1^2 = 1 \)

### Step 6: Substitute into the second part of the expression
Now substitute these values into the second part:
\[ 2 (cos^2 45° - sin^2 90°) = 2 \left(\frac{1}{2} - 1\right) = 2 \left(\frac{1}{2} - \frac{2}{2}\right) = 2 \left(-\frac{1}{2}\right) = -1 \]

### Step 7: Combine both parts of the expression
Now combine both parts:
\[ \left(\frac{337}{144}\right) - (-1) = \frac{337}{144} + 1 = \frac{337}{144} + \frac{144}{144} = \frac{481}{144} \]

### Final Answer
Thus, the final answer is:
\[ \frac{481}{144} \]