If `tantheta=tan30^(@).tan60^(@)andtheta` is an acute angle ,then `2theta` is equal to

Options

A. `30^(@)`
B. `45^(@)`
C. `90^(@)`
D. `0^(@)`

Answer

To solve the problem where \( \tan \theta = \tan 30^\circ \cdot \tan 60^\circ \) and \( \theta \) is an acute angle, we can follow these steps:

### Step 1: Calculate \( \tan 30^\circ \) and \( \tan 60^\circ \)
We know from trigonometric values:
- \( \tan 30^\circ = \frac{1}{\sqrt{3}} \)
- \( \tan 60^\circ = \sqrt{3} \)

### Step 2: Substitute the values into the equation
Now, substitute these values into the equation:
\[ \tan \theta = \tan 30^\circ \cdot \tan 60^\circ = \left(\frac{1}{\sqrt{3}}\right) \cdot (\sqrt{3}) \]

### Step 3: Simplify the right-hand side
Simplifying the right-hand side:
\[ \tan \theta = \frac{1}{\sqrt{3}} \cdot \sqrt{3} = 1 \]

### Step 4: Determine the angle \( \theta \)
Since \( \tan \theta = 1 \), we know that:
\[ \theta = 45^\circ \]
because \( \tan 45^\circ = 1 \).

### Step 5: Calculate \( 2\theta \)
Now, we need to find \( 2\theta \):
\[ 2\theta = 2 \cdot 45^\circ = 90^\circ \]

### Final Answer
Thus, \( 2\theta \) is equal to \( 90^\circ \).

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