Find the vlaue of `4sin^(2)30^(@)+tan^(2)60^(@)+sec^(2)45^(@)`.

Answer

To find the value of \(4\sin^2 30^\circ + \tan^2 60^\circ + \sec^2 45^\circ\), we will evaluate each trigonometric function step by step.

### 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 value of \(\sin 30^\circ\):
\[ \sin^2 30^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 3: Calculate \(4\sin^2 30^\circ\)
Next, we multiply by 4:
\[ 4\sin^2 30^\circ = 4 \times \frac{1}{4} = 1 \]

### Step 4: Calculate \(\tan 60^\circ\)
We know that:
\[ \tan 60^\circ = \sqrt{3} \]

### Step 5: Calculate \(\tan^2 60^\circ\)
Now, we square the value of \(\tan 60^\circ\):
\[ \tan^2 60^\circ = (\sqrt{3})^2 = 3 \]

### Step 6: Calculate \(\sec 45^\circ\)
We know that:
\[ \sec 45^\circ = \frac{1}{\cos 45^\circ} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]

### Step 7: Calculate \(\sec^2 45^\circ\)
Now, we square the value of \(\sec 45^\circ\):
\[ \sec^2 45^\circ = (\sqrt{2})^2 = 2 \]

### Step 8: Combine all the values
Now we can combine all the calculated values:
\[ 4\sin^2 30^\circ + \tan^2 60^\circ + \sec^2 45^\circ = 1 + 3 + 2 \]

### Step 9: Final Calculation
Adding these values together:
\[ 1 + 3 + 2 = 6 \]

Thus, the final answer is:
\[ \boxed{6} \]