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

Answer

To evaluate the expression

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

we will first find the values of the trigonometric functions involved.

### Step 1: Calculate \( \cos 60^\circ \)

The value of \( \cos 60^\circ \) is

\[ \cos 60^\circ = \frac{1}{2} \]

### Step 2: Calculate \( \cos^2 60^\circ \)

Now, we square the value:

\[ \cos^2 60^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 3: Calculate \( \cos 30^\circ \)

The value of \( \cos 30^\circ \) is

\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]

### Step 4: Calculate \( \cos^2 30^\circ \)

Now, we square the value:

\[ \cos^2 30^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

### Step 5: Calculate \( \tan 45^\circ \)

The value of \( \tan 45^\circ \) is

\[ \tan 45^\circ = 1 \]

### Step 6: Calculate \( \tan^2 45^\circ \)

Now, we square the value:

\[ \tan^2 45^\circ = 1^2 = 1 \]

### Step 7: Calculate \( \sin 30^\circ \)

The value of \( \sin 30^\circ \) is

\[ \sin 30^\circ = \frac{1}{2} \]

### Step 8: Calculate \( \sin^2 30^\circ \)

Now, we square the value:

\[ \sin^2 30^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 9: Calculate \( \cos^2 60^\circ \) (already calculated)

We already know:

\[ \cos^2 60^\circ = \frac{1}{4} \]

### Step 10: Substitute values into the expression

Now we substitute all the calculated values into the original expression:

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

### Step 11: Simplify the numerator

Calculating the numerator:

\[ 5 \cdot \frac{1}{4} = \frac{5}{4} \]
\[ 4 \cdot \frac{3}{4} = 3 \]
So, the numerator becomes:

\[ \frac{5}{4} + 3 - 1 = \frac{5}{4} + \frac{12}{4} - \frac{4}{4} = \frac{5 + 12 - 4}{4} = \frac{13}{4} \]

### Step 12: Simplify the denominator

Calculating the denominator:

\[ \frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \]

### Step 13: Final calculation

Now we have:

\[ \frac{\frac{13}{4}}{\frac{1}{2}} = \frac{13}{4} \cdot \frac{2}{1} = \frac{26}{4} = \frac{13}{2} \]

### Final Answer

Thus, the value of the expression is

\[ \frac{13}{2} \]