The value of `("tan"^2 30^@ + 1/2 "sin"^2 90^@ + 1/8 "cot"^2 60^@ + "sin"^2 30^@ "cos"^2 45^@)/("sin" 60^@ "cos"30^@ - "cos"60^@ "sin"30^@)`

Options

A. `1 3/4`
B. 2
C. `2 1/2`
D. `3 1/2`

Answer

To solve the given expression step by step, we need to evaluate both the numerator and the denominator separately.

### Step 1: Evaluate the numerator
The numerator is given as:
\[ \tan^2 30^\circ + \frac{1}{2} \sin^2 90^\circ + \frac{1}{8} \cot^2 60^\circ + \sin^2 30^\circ \cos^2 45^\circ \]

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

2. **Calculate \(\sin^2 90^\circ\)**:
\[ \sin 90^\circ = 1 \quad \Rightarrow \quad \sin^2 90^\circ = 1^2 = 1 \]
Thus, \(\frac{1}{2} \sin^2 90^\circ = \frac{1}{2} \cdot 1 = \frac{1}{2}\).

3. **Calculate \(\cot^2 60^\circ\)**:
\[ \cot 60^\circ = \frac{1}{\tan 60^\circ} = \frac{1}{\sqrt{3}} \quad \Rightarrow \quad \cot^2 60^\circ = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3} \]
Thus, \(\frac{1}{8} \cot^2 60^\circ = \frac{1}{8} \cdot \frac{1}{3} = \frac{1}{24}\).

4. **Calculate \(\sin^2 30^\circ\) and \(\cos^2 45^\circ\)**:
\[ \sin 30^\circ = \frac{1}{2} \quad \Rightarrow \quad \sin^2 30^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \quad \Rightarrow \quad \cos^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]
Thus, \(\sin^2 30^\circ \cos^2 45^\circ = \frac{1}{4} \cdot \frac{1}{2} = \frac{1}{8}\).

5. **Combine all parts of the numerator**:
\[ \text{Numerator} = \frac{1}{3} + \frac{1}{2} + \frac{1}{24} + \frac{1}{8} \]

To add these fractions, we need a common denominator. The least common multiple of \(3, 2, 24, 8\) is \(24\).

- Convert each term:
\[ \frac{1}{3} = \frac{8}{24}, \quad \frac{1}{2} = \frac{12}{24}, \quad \frac{1}{24} = \frac{1}{24}, \quad \frac{1}{8} = \frac{3}{24} \]
- Now add them:
\[ \text{Numerator} = \frac{8 + 12 + 1 + 3}{24} = \frac{24}{24} = 1 \]

### Step 2: Evaluate the denominator
The denominator is given as:
\[ \sin 60^\circ \cos 30^\circ - \cos 60^\circ \sin 30^\circ \]

1. **Calculate \(\sin 60^\circ\) and \(\cos 30^\circ\)**:
\[ \sin 60^\circ = \frac{\sqrt{3}}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2} \]
Thus, \(\sin 60^\circ \cos 30^\circ = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{3}{4}\).

2. **Calculate \(\cos 60^\circ\) and \(\sin 30^\circ\)**:
\[ \cos 60^\circ = \frac{1}{2}, \quad \sin 30^\circ = \frac{1}{2} \]
Thus, \(\cos 60^\circ \sin 30^\circ = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}\).

3. **Combine the parts of the denominator**:
\[ \text{Denominator} = \frac{3}{4} - \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \]

### Step 3: Final calculation
Now we can compute the final value:
\[ \text{Value} = \frac{\text{Numerator}}{\text{Denominator}} = \frac{1}{\frac{1}{2}} = 2 \]

### Final Answer
The value of the expression is \(2\).