The numerical value of ` ( cos^2 45^@ ) /( sin ^2 60 ^@) +( cos ^2 60^@ )/( sin ^2 45 ^@ ) -(tan ^2 30 ^@ )/( cot^2 45^@ ) -( sin^2 30 ^@ )/( cot^2 30^@)` is

Options

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

Answer

To solve the given expression step by step, we will use trigonometric identities and values.

### Given Expression:
\[ \frac{\cos^2 45^\circ}{\sin^2 60^\circ} + \frac{\cos^2 60^\circ}{\sin^2 45^\circ} - \frac{\tan^2 30^\circ}{\cot^2 45^\circ} - \frac{\sin^2 30^\circ}{\cot^2 30^\circ} \]

### Step 1: Calculate the trigonometric values
- \(\cos 45^\circ = \frac{1}{\sqrt{2}}\)
- \(\sin 60^\circ = \frac{\sqrt{3}}{2}\)
- \(\cos 60^\circ = \frac{1}{2}\)
- \(\sin 45^\circ = \frac{1}{\sqrt{2}}\)
- \(\tan 30^\circ = \frac{1}{\sqrt{3}}\) so \(\tan^2 30^\circ = \frac{1}{3}\)
- \(\cot 45^\circ = 1\) so \(\cot^2 45^\circ = 1\)
- \(\sin 30^\circ = \frac{1}{2}\) so \(\sin^2 30^\circ = \frac{1}{4}\)
- \(\cot 30^\circ = \sqrt{3}\) so \(\cot^2 30^\circ = 3\)

### Step 2: Substitute the values into the expression
Now substituting these values into the expression:
\[ \frac{\left(\frac{1}{\sqrt{2}}\right)^2}{\left(\frac{\sqrt{3}}{2}\right)^2} + \frac{\left(\frac{1}{2}\right)^2}{\left(\frac{1}{\sqrt{2}}\right)^2} - \frac{\frac{1}{3}}{1} - \frac{\frac{1}{4}}{3} \]

### Step 3: Simplify each term
1. \(\frac{\left(\frac{1}{\sqrt{2}}\right)^2}{\left(\frac{\sqrt{3}}{2}\right)^2} = \frac{\frac{1}{2}}{\frac{3}{4}} = \frac{1}{2} \cdot \frac{4}{3} = \frac{2}{3}\)
2. \(\frac{\left(\frac{1}{2}\right)^2}{\left(\frac{1}{\sqrt{2}}\right)^2} = \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{4} \cdot 2 = \frac{1}{2}\)
3. \(-\frac{1}{3}\)
4. \(-\frac{\frac{1}{4}}{3} = -\frac{1}{12}\)

### Step 4: Combine the results
Now combining all the results:
\[ \frac{2}{3} + \frac{1}{2} - \frac{1}{3} - \frac{1}{12} \]

### Step 5: Find a common denominator
The common denominator for \(3, 2, 3, 12\) is \(12\):
- \(\frac{2}{3} = \frac{8}{12}\)
- \(\frac{1}{2} = \frac{6}{12}\)
- \(-\frac{1}{3} = -\frac{4}{12}\)
- \(-\frac{1}{12} = -\frac{1}{12}\)

### Step 6: Combine the fractions
Now, adding these fractions together:
\[ \frac{8}{12} + \frac{6}{12} - \frac{4}{12} - \frac{1}{12} = \frac{8 + 6 - 4 - 1}{12} = \frac{9}{12} = \frac{3}{4} \]

### Final Answer:
The numerical value of the expression is \(\frac{3}{4}\).