Find the value of `4(sin^(4)30^(@)+cos^(4)30^(@))-3(cos^(2)45^(@)+sin^(2)90^(@))`.

Options

A. `-(1)/(2)`
B. `-2`
C. `2`
D. `(1)/(2)`

Explanation

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

### Step 1: Find the values of the trigonometric functions.
- We know:
- \( \sin 30^\circ = \frac{1}{2} \)
- \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)
- \( \cos 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \sin 90^\circ = 1 \)

### Step 2: Calculate \( \sin^4 30^\circ \) and \( \cos^4 30^\circ \).
- \( \sin^4 30^\circ = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \)
- \( \cos^4 30^\circ = \left(\frac{\sqrt{3}}{2}\right)^4 = \frac{3^2}{2^4} = \frac{9}{16} \)

### Step 3: Substitute these values into the expression.
- Now, we substitute these values into the expression:
\[ 4\left(\frac{1}{16} + \frac{9}{16}\right) - 3(\cos^2 45^\circ + \sin^2 90^\circ) \]

### Step 4: Simplify the expression inside the brackets.
- Inside the brackets:
\[ \frac{1}{16} + \frac{9}{16} = \frac{10}{16} = \frac{5}{8} \]
- Thus, we have:
\[ 4\left(\frac{5}{8}\right) - 3(\cos^2 45^\circ + \sin^2 90^\circ) \]

### Step 5: Calculate \( \cos^2 45^\circ \) and \( \sin^2 90^\circ \).
- \( \cos^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \)
- \( \sin^2 90^\circ = 1^2 = 1 \)

### Step 6: Substitute these values into the expression.
- Now substitute:
\[ 4\left(\frac{5}{8}\right) - 3\left(\frac{1}{2} + 1\right) \]

### Step 7: Simplify the expression.
- First, calculate \( \frac{1}{2} + 1 = \frac{1}{2} + \frac{2}{2} = \frac{3}{2} \).
- Now substitute:
\[ 4\left(\frac{5}{8}\right) - 3\left(\frac{3}{2}\right) \]
- Calculate \( 4 \times \frac{5}{8} = \frac{20}{8} = \frac{5}{2} \).
- Calculate \( 3 \times \frac{3}{2} = \frac{9}{2} \).

### Step 8: Final calculation.
- Now we have:
\[ \frac{5}{2} - \frac{9}{2} = \frac{5 - 9}{2} = \frac{-4}{2} = -2 \]

### Final Answer:
The value of the expression is \( -2 \).

---