Evaluate: `sin^(2)60^(@) cos^(2)45^(@) cos^(2)60^(@)" cosec"^(2)90^(@)`

Answer

To evaluate the expression \( \sin^2 60^\circ \cdot \cos^2 45^\circ \cdot \cos^2 60^\circ \cdot \csc^2 90^\circ \), we will follow these steps:

### Step 1: Calculate \( \sin 60^\circ \)
The value of \( \sin 60^\circ \) is known to be:
\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \]

### Step 2: Calculate \( \sin^2 60^\circ \)
Now we square the value:
\[ \sin^2 60^\circ = \left( \frac{\sqrt{3}}{2} \right)^2 = \frac{3}{4} \]

### Step 3: Calculate \( \cos 45^\circ \)
The value of \( \cos 45^\circ \) is:
\[ \cos 45^\circ = \frac{\sqrt{2}}{2} \]

### Step 4: Calculate \( \cos^2 45^\circ \)
Now we square the value:
\[ \cos^2 45^\circ = \left( \frac{\sqrt{2}}{2} \right)^2 = \frac{2}{4} = \frac{1}{2} \]

### Step 5: Calculate \( \cos 60^\circ \)
The value of \( \cos 60^\circ \) is:
\[ \cos 60^\circ = \frac{1}{2} \]

### Step 6: Calculate \( \cos^2 60^\circ \)
Now we square the value:
\[ \cos^2 60^\circ = \left( \frac{1}{2} \right)^2 = \frac{1}{4} \]

### Step 7: Calculate \( \csc 90^\circ \)
The value of \( \csc 90^\circ \) is:
\[ \csc 90^\circ = \frac{1}{\sin 90^\circ} = \frac{1}{1} = 1 \]

### Step 8: Calculate \( \csc^2 90^\circ \)
Now we square the value:
\[ \csc^2 90^\circ = 1^2 = 1 \]

### Step 9: Combine all the values
Now we can substitute all the calculated values back into the original expression:
\[ \sin^2 60^\circ \cdot \cos^2 45^\circ \cdot \cos^2 60^\circ \cdot \csc^2 90^\circ = \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{1}{4} \cdot 1 \]

### Step 10: Perform the multiplication
Now we multiply these fractions:
\[ \frac{3}{4} \cdot \frac{1}{2} = \frac{3}{8} \]
\[ \frac{3}{8} \cdot \frac{1}{4} = \frac{3}{32} \]
\[ \frac{3}{32} \cdot 1 = \frac{3}{32} \]

### Final Answer
Thus, the final answer is:
\[ \frac{3}{32} \]