`(5sin^(2)30^(@)+cos^(2)45^(@)-4tan^(2)30^(@))/(2sin30^(@).cos30^(@)+tan45^(@))=?`

Options

A. `(5)/(6)(2-sqrt(3))`
B. `(5)/(6)(2+sqrt(3))`
C. `2-sqrt(3)`
D. `(1)/(6)(2-sqrt(3))`

Answer

To solve the expression

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

we will first substitute the known values of the trigonometric functions.

### Step 1: Substitute the values of trigonometric functions

- \(\sin 30^\circ = \frac{1}{2}\)
- \(\cos 30^\circ = \frac{\sqrt{3}}{2}\)
- \(\cos 45^\circ = \frac{1}{\sqrt{2}}\)
- \(\tan 30^\circ = \frac{1}{\sqrt{3}}\)
- \(\tan 45^\circ = 1\)

Substituting these values into the expression gives:

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

### Step 2: Simplify the numerator

Calculating each term in the numerator:

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

Now, substituting these values back into the numerator:

\[ \frac{5}{4} + \frac{1}{2} - \frac{4}{3} \]

### Step 3: Find a common denominator for the numerator

The common denominator for \(4\), \(2\), and \(3\) is \(12\). Now we convert each term:

1. \(\frac{5}{4} = \frac{15}{12}\)
2. \(\frac{1}{2} = \frac{6}{12}\)
3. \(\frac{4}{3} = \frac{16}{12}\)

Now substituting these values gives:

\[ \frac{15}{12} + \frac{6}{12} - \frac{16}{12} = \frac{15 + 6 - 16}{12} = \frac{5}{12} \]

### Step 4: Simplify the denominator

Now simplifying the denominator:

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

### Step 5: Combine the results

Now we have:

\[ \frac{\frac{5}{12}}{\frac{\sqrt{3}}{2} + 1} \]

### Step 6: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{5}{12} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{5(2 - \sqrt{3})}{12\left((\sqrt{3}/2) + 1\right)(2 - \sqrt{3})} \]

Calculating the denominator:

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

Thus, the final result is:

\[ \frac{5(2 - \sqrt{3})}{12} \]

### Final Answer

\[ \frac{5(2 - \sqrt{3})}{12} \]