What is the value of `4(sin^4 30^@ + cos^4 60^@) -3(cos^2 45^@ - sin^2 90^@)` ?

Answer

To solve the expression \( 4(\sin^4 30^\circ + \cos^4 60^\circ) - 3(\cos^2 45^\circ - \sin^2 90^\circ) \), we will follow these steps:

### Step 1: Calculate \(\sin 30^\circ\) and \(\cos 60^\circ\)
- We know that:
\[ \sin 30^\circ = \frac{1}{2} \]
\[ \cos 60^\circ = \frac{1}{2} \]

### Step 2: Calculate \(\sin^4 30^\circ\) and \(\cos^4 60^\circ\)
- Now, we can calculate:
\[ \sin^4 30^\circ = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]
\[ \cos^4 60^\circ = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]

### Step 3: Substitute values into the expression
- Substitute these values into the expression:
\[ 4\left(\frac{1}{16} + \frac{1}{16}\right) - 3(\cos^2 45^\circ - \sin^2 90^\circ) \]

### Step 4: Simplify the first part
- Simplifying the first part:
\[ 4\left(\frac{1}{16} + \frac{1}{16}\right) = 4\left(\frac{2}{16}\right) = 4\left(\frac{1}{8}\right) = \frac{4}{8} = \frac{1}{2} \]

### Step 5: Calculate \(\cos 45^\circ\) and \(\sin 90^\circ\)
- We know that:
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \quad \text{and} \quad \sin 90^\circ = 1 \]
- Therefore:
\[ \cos^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]
\[ \sin^2 90^\circ = 1^2 = 1 \]

### Step 6: Substitute values into the second part
- Substitute these values into the second part:
\[ -3\left(\frac{1}{2} - 1\right) = -3\left(-\frac{1}{2}\right) = \frac{3}{2} \]

### Step 7: Combine both parts
- Now, combine both parts:
\[ \frac{1}{2} + \frac{3}{2} = \frac{4}{2} = 2 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{2} \]