Find the value of :
`(5sin^(3)30^(@)+cos^(2)45^(@)-4tan^(2)30^(@))/(2sin 30^(@).cos30^(@)+tan45^(@))+cos 0^(@)`

Answer

To solve the expression

\[ \frac{5\sin^3(30^\circ) + \cos^2(45^\circ) - 4\tan^2(30^\circ)}{2\sin(30^\circ)\cos(30^\circ) + \tan(45^\circ)} + \cos(0^\circ), \]

we will evaluate each trigonometric function step by step.

### Step 1: Calculate the trigonometric values
1. **Calculate \(\sin(30^\circ)\)**:
\[ \sin(30^\circ) = \frac{1}{2} \]

2. **Calculate \(\cos(45^\circ)\)**:
\[ \cos(45^\circ) = \frac{\sqrt{2}}{2} \]

3. **Calculate \(\tan(30^\circ)\)**:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]

4. **Calculate \(\tan(45^\circ)\)**:
\[ \tan(45^\circ) = 1 \]

5. **Calculate \(\cos(0^\circ)\)**:
\[ \cos(0^\circ) = 1 \]

### Step 2: Substitute the values into the expression
Now substitute these values into the expression:

1. **Calculate \(\sin^3(30^\circ)\)**:
\[ \sin^3(30^\circ) = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]

2. **Calculate \(\cos^2(45^\circ)\)**:
\[ \cos^2(45^\circ) = \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{2}{4} = \frac{1}{2} \]

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

### Step 3: Substitute into the numerator
Now substitute these values into the numerator:

\[ 5\sin^3(30^\circ) + \cos^2(45^\circ) - 4\tan^2(30^\circ) = 5\left(\frac{1}{8}\right) + \frac{1}{2} - 4\left(\frac{1}{3}\right) \]

Calculating each term:

1. \(5\left(\frac{1}{8}\right) = \frac{5}{8}\)
2. \(\frac{1}{2} = \frac{4}{8}\)
3. \(4\left(\frac{1}{3}\right) = \frac{4}{3}\)

Now, find a common denominator to combine these fractions. The least common multiple of 8 and 3 is 24.

Convert each term:

1. \(\frac{5}{8} = \frac{15}{24}\)
2. \(\frac{4}{8} = \frac{12}{24}\)
3. \(\frac{4}{3} = \frac{32}{24}\)

Now substitute back into the numerator:

\[ \frac{15}{24} + \frac{12}{24} - \frac{32}{24} = \frac{15 + 12 - 32}{24} = \frac{-5}{24} \]

### Step 4: Substitute into the denominator
Now calculate the denominator:

\[ 2\sin(30^\circ)\cos(30^\circ) + \tan(45^\circ) = 2\left(\frac{1}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + 1 \]

Calculating:

1. \(2\left(\frac{1}{2}\right)\left(\frac{\sqrt{3}}{2}\right) = \frac{\sqrt{3}}{2}\)
2. \(1 = \frac{2}{2}\)

Now combine:

\[ \frac{\sqrt{3}}{2} + \frac{2}{2} = \frac{\sqrt{3} + 2}{2} \]

### Step 5: Combine the entire expression
Now substitute the numerator and denominator back into the expression:

\[ \frac{\frac{-5}{24}}{\frac{\sqrt{3} + 2}{2}} + 1 \]

This simplifies to:

\[ \frac{-5}{24} \cdot \frac{2}{\sqrt{3} + 2} + 1 = \frac{-10}{24(\sqrt{3} + 2)} + 1 \]

### Step 6: Simplify the expression
Now simplify:

\[ \frac{-10}{24(\sqrt{3} + 2)} + 1 = 1 - \frac{10}{24(\sqrt{3} + 2)} \]

To combine these, find a common denominator:

\[ \frac{24(\sqrt{3} + 2)}{24(\sqrt{3} + 2)} - \frac{10}{24(\sqrt{3} + 2)} = \frac{24(\sqrt{3} + 2) - 10}{24(\sqrt{3} + 2)} \]

This simplifies to:

\[ \frac{24\sqrt{3} + 48 - 10}{24(\sqrt{3} + 2)} = \frac{24\sqrt{3} + 38}{24(\sqrt{3} + 2)} \]

### Final Answer
Thus, the final value of the expression is:

\[ \frac{24\sqrt{3} + 38}{24(\sqrt{3} + 2)} \]