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

Answer

To find the value of \( \tan^2 30^\circ + \tan^2 45^\circ + \tan^2 60^\circ \), we can follow these steps:

### Step 1: Identify the values of the tangent for the standard angles.
We know the following values:
- \( \tan 30^\circ = \frac{1}{\sqrt{3}} \)
- \( \tan 45^\circ = 1 \)
- \( \tan 60^\circ = \sqrt{3} \)

### Step 2: Square each of the tangent values.
Now we will square each of these values:
- \( \tan^2 30^\circ = \left( \frac{1}{\sqrt{3}} \right)^2 = \frac{1}{3} \)
- \( \tan^2 45^\circ = 1^2 = 1 \)
- \( \tan^2 60^\circ = (\sqrt{3})^2 = 3 \)

### Step 3: Add the squared values together.
Now we will add these squared values:
\[ \tan^2 30^\circ + \tan^2 45^\circ + \tan^2 60^\circ = \frac{1}{3} + 1 + 3 \]

### Step 4: Convert all terms to have a common denominator.
To add these fractions, we can convert them to have a common denominator of 3:
- \( 1 = \frac{3}{3} \)
- \( 3 = \frac{9}{3} \)

Now we can rewrite the equation:
\[ \tan^2 30^\circ + \tan^2 45^\circ + \tan^2 60^\circ = \frac{1}{3} + \frac{3}{3} + \frac{9}{3} \]

### Step 5: Combine the fractions.
Now we can combine the fractions:
\[ \tan^2 30^\circ + \tan^2 45^\circ + \tan^2 60^\circ = \frac{1 + 3 + 9}{3} = \frac{13}{3} \]

### Final Answer:
Thus, the value of \( \tan^2 30^\circ + \tan^2 45^\circ + \tan^2 60^\circ \) is \( \frac{13}{3} \).

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