Find the value of :
`cos^(2)60^(@)+sec^(2)30^(@)+tan^(2)45^(@)`

Answer

To find the value of the expression \( \cos^2(60^\circ) + \sec^2(30^\circ) + \tan^2(45^\circ) \), we will calculate each term step by step.

### Step 1: Calculate \( \cos^2(60^\circ) \)
We know that:
\[ \cos(60^\circ) = \frac{1}{2} \]
Thus,
\[ \cos^2(60^\circ) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 2: Calculate \( \sec^2(30^\circ) \)
The secant function is the reciprocal of the cosine function:
\[ \sec(30^\circ) = \frac{1}{\cos(30^\circ)} \]
We know that:
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
Therefore,
\[ \sec(30^\circ) = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]
Now, we calculate \( \sec^2(30^\circ) \):
\[ \sec^2(30^\circ) = \left(\frac{2}{\sqrt{3}}\right)^2 = \frac{4}{3} \]

### Step 3: Calculate \( \tan^2(45^\circ) \)
We know that:
\[ \tan(45^\circ) = 1 \]
Thus,
\[ \tan^2(45^\circ) = 1^2 = 1 \]

### Step 4: Combine all the values
Now, we can combine all the calculated values:
\[ \cos^2(60^\circ) + \sec^2(30^\circ) + \tan^2(45^\circ) = \frac{1}{4} + \frac{4}{3} + 1 \]

### Step 5: Find a common denominator
The common denominator for \( 4 \), \( 3 \), and \( 1 \) is \( 12 \):
\[ \frac{1}{4} = \frac{3}{12}, \quad \frac{4}{3} = \frac{16}{12}, \quad 1 = \frac{12}{12} \]
Now, we can add these fractions:
\[ \frac{3}{12} + \frac{16}{12} + \frac{12}{12} = \frac{3 + 16 + 12}{12} = \frac{31}{12} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{31}{12} \]