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

Options

A. `2 (5)/(16)`
B. `(67)/(12)`
C. 0
D. 1

Answer

To evaluate the expression

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

we will follow these steps:

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

### Step 2: Substitute the values into the expression
Substituting the values we calculated:

\[ \cos^2 60^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
\[ \sec^2 30^\circ = \left(\frac{2}{\sqrt{3}}\right)^2 = \frac{4}{3} \]
\[ \tan^2 45^\circ = 1^2 = 1 \]
\[ \sin^2 30^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
\[ \cos^2 30^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

### Step 3: Substitute these values into the expression
Now substituting these values into the expression:

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

### Step 4: Simplify the numerator
Calculating the numerator:

\[ 5 \cdot \frac{1}{4} = \frac{5}{4} \]
\[ 4 \cdot \frac{4}{3} = \frac{16}{3} \]
So, the numerator becomes:

\[ \frac{5}{4} + \frac{16}{3} - 1 \]

To combine these, we need a common denominator. The least common multiple of 4 and 3 is 12.

Converting each term:
\[ \frac{5}{4} = \frac{15}{12}, \quad \frac{16}{3} = \frac{64}{12}, \quad 1 = \frac{12}{12} \]

Now substituting back into the numerator:

\[ \frac{15}{12} + \frac{64}{12} - \frac{12}{12} = \frac{15 + 64 - 12}{12} = \frac{67}{12} \]

### Step 5: Simplify the denominator
Now for the denominator:

\[ \frac{1}{4} + \frac{3}{4} = 1 \]

### Step 6: Final Calculation
Now substituting back into the expression:

\[ \frac{\frac{67}{12}}{1} = \frac{67}{12} \]

Thus, the final answer is:

\[ \frac{67}{12} \]