Find the value of :
`sin^(2)30^(@)+cos^(2)30^(@)+cot^(2)45^(@)`

Answer

To find the value of \( \sin^2(30^\circ) + \cos^2(30^\circ) + \cot^2(45^\circ) \), we will follow these steps:

### Step 1: Calculate \( \sin(30^\circ) \)
We know that:
\[ \sin(30^\circ) = \frac{1}{2} \]

### Step 2: Calculate \( \sin^2(30^\circ) \)
Now, we square the sine value:
\[ \sin^2(30^\circ) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 3: Calculate \( \cos(30^\circ) \)
Next, we find the cosine value:
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]

### Step 4: Calculate \( \cos^2(30^\circ) \)
Now, we square the cosine value:
\[ \cos^2(30^\circ) = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

### Step 5: Calculate \( \cot(45^\circ) \)
We know that:
\[ \cot(45^\circ) = 1 \]

### Step 6: Calculate \( \cot^2(45^\circ) \)
Now, we square the cotangent value:
\[ \cot^2(45^\circ) = 1^2 = 1 \]

### Step 7: Combine all the values
Now we can add all the squared values together:
\[ \sin^2(30^\circ) + \cos^2(30^\circ) + \cot^2(45^\circ) = \frac{1}{4} + \frac{3}{4} + 1 \]

### Step 8: Simplify the expression
First, we add \( \frac{1}{4} \) and \( \frac{3}{4} \):
\[ \frac{1}{4} + \frac{3}{4} = \frac{4}{4} = 1 \]
Now, add this to \( 1 \):
\[ 1 + 1 = 2 \]

### Final Answer
Thus, the value of \( \sin^2(30^\circ) + \cos^2(30^\circ) + \cot^2(45^\circ) \) is:
\[ \boxed{2} \]