The value of `(1)/(2) sin ^(2) 90^(@) sin^(2) 30^(@) cos ^(2) 45^(@) + 4 tan ^(2) 30^(@) + (1)/(2) sin^(2) 90^(@) - 2 cos ^(2) 90^(@) ` is :

Options

A. `(45)/(24)`
B. `(46)/(24)`
C. `(47)/(24)`
D. `(49)/(24)`

Answer

To solve the expression

\[ \frac{1}{2} \sin^2 90^\circ \sin^2 30^\circ \cos^2 45^\circ + 4 \tan^2 30^\circ + \frac{1}{2} \sin^2 90^\circ - 2 \cos^2 90^\circ, \]

we will evaluate each trigonometric function step by step.

### Step 1: Calculate \(\sin^2 90^\circ\)
\[ \sin 90^\circ = 1 \implies \sin^2 90^\circ = 1^2 = 1. \]

### Step 2: Calculate \(\sin^2 30^\circ\)
\[ \sin 30^\circ = \frac{1}{2} \implies \sin^2 30^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4}. \]

### Step 3: Calculate \(\cos^2 45^\circ\)
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \implies \cos^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}. \]

### Step 4: Calculate \(\tan^2 30^\circ\)
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \implies \tan^2 30^\circ = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3}. \]

### Step 5: Calculate \(\cos^2 90^\circ\)
\[ \cos 90^\circ = 0 \implies \cos^2 90^\circ = 0^2 = 0. \]

### Step 6: Substitute values into the expression
Now we substitute the calculated values back into the expression:

\[ \frac{1}{2} \cdot 1 \cdot \frac{1}{4} \cdot \frac{1}{2} + 4 \cdot \frac{1}{3} + \frac{1}{2} \cdot 1 - 2 \cdot 0. \]

### Step 7: Simplify the expression
1. First term:
\[ \frac{1}{2} \cdot 1 \cdot \frac{1}{4} \cdot \frac{1}{2} = \frac{1}{16}. \]

2. Second term:
\[ 4 \cdot \frac{1}{3} = \frac{4}{3}. \]

3. Third term:
\[ \frac{1}{2} \cdot 1 = \frac{1}{2}. \]

4. Fourth term:
\[ -2 \cdot 0 = 0. \]

Combining all these:
\[ \frac{1}{16} + \frac{4}{3} + \frac{1}{2}. \]

### Step 8: Find a common denominator
The least common multiple (LCM) of 16, 3, and 2 is 48. We convert each term:
1. \(\frac{1}{16} = \frac{3}{48}\),
2. \(\frac{4}{3} = \frac{64}{48}\),
3. \(\frac{1}{2} = \frac{24}{48}\).

### Step 9: Combine the fractions
Now, we add these fractions:
\[ \frac{3}{48} + \frac{64}{48} + \frac{24}{48} = \frac{3 + 64 + 24}{48} = \frac{91}{48}. \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{91}{48}. \]

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