Evaluate : `(5 sin ^(2) 30^(@) + cos ^(2) 45^(@) + 4 tan ^(2) 60^(@))/( 2 sin 30^(@) cos 60^(@) + tan 45^(@))`

Options

A. 1
B. `9 (1)/(6)`
C. ` 7 (3)/(7)`
D. `(47)/(12)`

Answer

To evaluate the expression

\[ \frac{5 \sin^2 30^\circ + \cos^2 45^\circ + 4 \tan^2 60^\circ}{2 \sin 30^\circ \cos 60^\circ + \tan 45^\circ} \]

we will follow these steps:

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

### Step 2: Substitute the values into the expression
Substituting these values into the expression gives:

\[ \frac{5 \left(\frac{1}{2}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 + 4 \left(\sqrt{3}\right)^2}{2 \left(\frac{1}{2}\right) \left(\frac{1}{2}\right) + 1} \]

### Step 3: Simplify the numerator
Calculating each term in the numerator:

- \(5 \left(\frac{1}{2}\right)^2 = 5 \cdot \frac{1}{4} = \frac{5}{4}\)
- \(\left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}\)
- \(4 \left(\sqrt{3}\right)^2 = 4 \cdot 3 = 12\)

Now, adding these values together:

\[ \frac{5}{4} + \frac{1}{2} + 12 \]

To add these fractions, we need a common denominator. The least common multiple of 4 and 2 is 4. Thus, we rewrite \(\frac{1}{2}\) as \(\frac{2}{4}\):

\[ \frac{5}{4} + \frac{2}{4} + 12 = \frac{5 + 2}{4} + 12 = \frac{7}{4} + 12 \]

To add \(12\) (which is \(\frac{48}{4}\)):

\[ \frac{7}{4} + \frac{48}{4} = \frac{55}{4} \]

### Step 4: Simplify the denominator
Now, simplifying the denominator:

\[ 2 \left(\frac{1}{2}\right) \left(\frac{1}{2}\right) + 1 = 2 \cdot \frac{1}{4} + 1 = \frac{1}{2} + 1 = \frac{1}{2} + \frac{2}{2} = \frac{3}{2} \]

### Step 5: Combine the results
Now, we combine the results from the numerator and denominator:

\[ \frac{\frac{55}{4}}{\frac{3}{2}} = \frac{55}{4} \cdot \frac{2}{3} = \frac{55 \cdot 2}{4 \cdot 3} = \frac{110}{12} = \frac{55}{6} \]

### Step 6: Convert to mixed fraction
Now, we convert \(\frac{55}{6}\) into a mixed fraction:

Dividing \(55\) by \(6\):

- \(6\) goes into \(55\) \(9\) times (since \(6 \times 9 = 54\)), leaving a remainder of \(1\).

Thus, we can write:

\[ \frac{55}{6} = 9 \frac{1}{6} \]

### Final Answer:
The final answer is

\[ 9 \frac{1}{6} \]