Choose the correct answer in each of the following questions:
`sin^(2)30^(@)+4cot^(2)45^(@)-sec^(2)60^(@)=?`

Options

A. 0
B. `(1)/(4)`
C. `4`
D. 1

Explanation

To solve the expression \( \sin^2 30^\circ + 4 \cot^2 45^\circ - \sec^2 60^\circ \), we will evaluate each trigonometric function step by step.

### Step 1: Calculate \( \sin^2 30^\circ \)
We know from trigonometric values that:
\[ \sin 30^\circ = \frac{1}{2} \]
Thus,
\[ \sin^2 30^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 2: Calculate \( \cot^2 45^\circ \)
The cotangent of 45 degrees is:
\[ \cot 45^\circ = 1 \]
Therefore,
\[ \cot^2 45^\circ = 1^2 = 1 \]
Now, we multiply this by 4:
\[ 4 \cot^2 45^\circ = 4 \times 1 = 4 \]

### Step 3: Calculate \( \sec^2 60^\circ \)
The secant of 60 degrees is:
\[ \sec 60^\circ = \frac{1}{\cos 60^\circ} \]
Since \( \cos 60^\circ = \frac{1}{2} \), we have:
\[ \sec 60^\circ = \frac{1}{\frac{1}{2}} = 2 \]
Thus,
\[ \sec^2 60^\circ = 2^2 = 4 \]

### Step 4: Substitute the values into the expression
Now we substitute the calculated values into the original expression:
\[ \sin^2 30^\circ + 4 \cot^2 45^\circ - \sec^2 60^\circ = \frac{1}{4} + 4 - 4 \]

### Step 5: Simplify the expression
Now we simplify:
\[ \frac{1}{4} + 4 - 4 = \frac{1}{4} + 0 = \frac{1}{4} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{1}{4} \]