What is the value of `tan30^@ + tan60^@`?

Options

A. `(4sqrt3)/3`
B. `(2sqrt3)/alpha`
C. `(sqrt3)/alpha`
D. `1/2`

Answer

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

### Step 1: Find \( \tan 30^\circ \)
The value of \( \tan 30^\circ \) is known to be:
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \quad \text{(or equivalently, } \frac{\sqrt{3}}{3}\text{)} \]

### Step 2: Find \( \tan 60^\circ \)
The value of \( \tan 60^\circ \) is known to be:
\[ \tan 60^\circ = \sqrt{3} \]

### Step 3: Add \( \tan 30^\circ \) and \( \tan 60^\circ \)
Now, we can add the two values:
\[ \tan 30^\circ + \tan 60^\circ = \frac{1}{\sqrt{3}} + \sqrt{3} \]

### Step 4: Find a common denominator
To add these fractions, we need a common denominator. The common denominator is \( \sqrt{3} \):
\[ \tan 30^\circ + \tan 60^\circ = \frac{1}{\sqrt{3}} + \frac{3}{\sqrt{3}} = \frac{1 + 3}{\sqrt{3}} = \frac{4}{\sqrt{3}} \]

### Step 5: Simplify the result
We can simplify \( \frac{4}{\sqrt{3}} \) by rationalizing the denominator:
\[ \frac{4}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{4\sqrt{3}}{3} \]

### Final Answer
Thus, the value of \( \tan 30^\circ + \tan 60^\circ \) is:
\[ \frac{4\sqrt{3}}{3} \]

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