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 \( \sin(60°) \)
The value of \( \sin(60°) \) is \( \frac{\sqrt{3}}{2} \).

### Step 2: Calculate \( \sin^4(60°) \)
Now, we find \( \sin^4(60°) \):
\[ \sin^4(60°) = \left(\frac{\sqrt{3}}{2}\right)^4 = \frac{3^2}{2^4} = \frac{9}{16} \]

### Step 3: Calculate \( \sec(30°) \)
The value of \( \sec(30°) \) is the reciprocal of \( \cos(30°) \). Since \( \cos(30°) = \frac{\sqrt{3}}{2} \), we have:
\[ \sec(30°) = \frac{1}{\cos(30°)} = \frac{2}{\sqrt{3}} \]

### Step 4: Calculate \( \sec^4(30°) \)
Now, we find \( \sec^4(30°) \):
\[ \sec^4(30°) = \left(\frac{2}{\sqrt{3}}\right)^4 = \frac{2^4}{3^2} = \frac{16}{9} \]

### Step 5: Combine \( \sin^4(60°) \) and \( \sec^4(30°) \)
Now we add \( \sin^4(60°) \) and \( \sec^4(30°) \):
\[ \sin^4(60°) + \sec^4(30°) = \frac{9}{16} + \frac{16}{9} \]
To add these fractions, we need a common denominator. The least common multiple of 16 and 9 is 144:
\[ \frac{9}{16} = \frac{9 \times 9}{16 \times 9} = \frac{81}{144} \]
\[ \frac{16}{9} = \frac{16 \times 16}{9 \times 16} = \frac{256}{144} \]
Now, adding these:
\[ \frac{81}{144} + \frac{256}{144} = \frac{337}{144} \]

### Step 6: Calculate \( \cos(45°) \)
The value of \( \cos(45°) \) is \( \frac{1}{\sqrt{2}} \).

### Step 7: Calculate \( \cos^2(45°) \)
Now, we find \( \cos^2(45°) \):
\[ \cos^2(45°) = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

### Step 8: Calculate \( \sin(90°) \)
The value of \( \sin(90°) \) is 1.

### Step 9: Calculate \( \sin^2(90°) \)
Now, we find \( \sin^2(90°) \):
\[ \sin^2(90°) = 1^2 = 1 \]

### Step 10: Combine \( 2(cos^2(45°) - sin^2(90°)) \)
Now we calculate:
\[ 2(cos^2(45°) - sin^2(90°)) = 2\left(\frac{1}{2} - 1\right) = 2\left(\frac{1}{2} - \frac{2}{2}\right) = 2\left(-\frac{1}{2}\right) = -1 \]

### Step 11: Final Calculation
Now we substitute back into the original expression:
\[ \left(\frac{337}{144}\right) - (-1) = \frac{337}{144} + 1 = \frac{337}{144} + \frac{144}{144} = \frac{481}{144} \]

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