Evaluate `[4(sin^(2)30^(@)+cos^(4)60^(@))-3(cos^(2)45^(@)-sin^(2)90^(@))]xx(2cos^(2)60^(@)+3sec^(2)30^(@)-2tan^(2)45^(@))/(sin^(2)30^(@)+cos^(2)45^(@))`

Options

A. `55/6`
B. 0
C. 1
D. `32/3`

Answer

To evaluate the expression

\[ \frac{[4(\sin^2 30^\circ + \cos^4 60^\circ) - 3(\cos^2 45^\circ - \sin^2 90^\circ)] \times [2\cos^2 60^\circ + 3\sec^2 30^\circ - 2\tan^2 45^\circ]}{\sin^2 30^\circ + \cos^2 45^\circ}, \]

we will follow these steps:

### Step 1: Calculate the trigonometric values
- \(\sin 30^\circ = \frac{1}{2}\)
- \(\cos 60^\circ = \frac{1}{2}\)
- \(\cos 45^\circ = \frac{1}{\sqrt{2}}\)
- \(\sin 90^\circ = 1\)
- \(\sec 30^\circ = \frac{1}{\cos 30^\circ} = \frac{2}{\sqrt{3}}\)
- \(\tan 45^\circ = 1\)

### Step 2: Substitute the values into the expression
Substituting the values we found into the expression gives us:

\[ = \frac{[4\left(\left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^4\right) - 3\left(\left(\frac{1}{\sqrt{2}}\right)^2 - 1\right)] \times [2\left(\frac{1}{2}\right)^2 + 3\left(\frac{2}{\sqrt{3}}\right)^2 - 2(1)]}{\left(\frac{1}{2}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2} \]

### Step 3: Simplify the components
1. **Numerator Part 1:**
- \(\sin^2 30^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4}\)
- \(\cos^4 60^\circ = \left(\frac{1}{2}\right)^4 = \frac{1}{16}\)
- \(\cos^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}\)

Thus,
\[ 4\left(\frac{1}{4} + \frac{1}{16}\right) - 3\left(\frac{1}{2} - 1\right) = 4\left(\frac{4 + 1}{16}\right) - 3\left(-\frac{1}{2}\right) \]
\[ = 4 \cdot \frac{5}{16} + \frac{3}{2} = \frac{20}{16} + \frac{24}{16} = \frac{44}{16} = \frac{11}{4} \]

2. **Numerator Part 2:**
- \(\cos^2 60^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4}\)
- \(\sec^2 30^\circ = \left(\frac{2}{\sqrt{3}}\right)^2 = \frac{4}{3}\)
- Thus,
\[ 2\left(\frac{1}{4}\right) + 3\left(\frac{4}{3}\right) - 2(1) = \frac{1}{2} + 4 - 2 = \frac{1}{2} + 2 = \frac{5}{2} \]

### Step 4: Combine the results
Now we combine the results from the numerator:

\[ \frac{11}{4} \times \frac{5}{2} = \frac{55}{8} \]

### Step 5: Denominator
- \(\sin^2 30^\circ + \cos^2 45^\circ = \frac{1}{4} + \frac{1}{2} = \frac{1}{4} + \frac{2}{4} = \frac{3}{4}\)

### Step 6: Final Calculation
Now we can write the final expression:

\[ \frac{\frac{55}{8}}{\frac{3}{4}} = \frac{55}{8} \times \frac{4}{3} = \frac{55 \times 4}{8 \times 3} = \frac{220}{24} = \frac{55}{6} \]

Thus, the final answer is:

\[ \boxed{\frac{55}{6}} \]