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

Answer

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

We know that:
\[ \sec 30^\circ = \frac{1}{\cos 30^\circ} \]
And since \( \cos 30^\circ = \frac{\sqrt{3}}{2} \), we have:
\[ \sec 30^\circ = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]
Thus,
\[ \sec^2 30^\circ = \left(\frac{2}{\sqrt{3}}\right)^2 = \frac{4}{3} \]

### 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 the results

Now we can combine all the results:
\[ \cos^2 60^\circ + \sec^2 30^\circ + \tan^2 45^\circ = \frac{1}{4} + \frac{4}{3} + 1 \]

### Step 5: Find a common denominator

The common denominator for \( 4 \), \( 3 \), and \( 1 \) is \( 12 \). We convert each term:
\[ \frac{1}{4} = \frac{3}{12}, \quad \frac{4}{3} = \frac{16}{12}, \quad 1 = \frac{12}{12} \]

### Step 6: Add the fractions

Now we add the fractions:
\[ \frac{3}{12} + \frac{16}{12} + \frac{12}{12} = \frac{3 + 16 + 12}{12} = \frac{31}{12} \]

### Final Answer

Thus, the final value is:
\[ \cos^2 60^\circ + \sec^2 30^\circ + \tan^2 45^\circ = \frac{31}{12} \]

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