If `tanx= 3cotx " then " x=?`

Options

A. `45^(@)`
B. `60^(@)`
C. `30^(@)`
D. `15^(@)`

Explanation

To solve the equation \( \tan x = 3 \cot x \), we will follow these steps:

### Step 1: Rewrite the equation using trigonometric identities
We know that \( \cot x = \frac{1}{\tan x} \). Therefore, we can rewrite the equation as:
\[ \tan x = 3 \cdot \frac{1}{\tan x} \]

### Step 2: Multiply both sides by \( \tan x \)
To eliminate the fraction, we multiply both sides by \( \tan x \) (assuming \( \tan x \neq 0 \)):
\[ \tan^2 x = 3 \]

### Step 3: Solve for \( \tan x \)
Now, we take the square root of both sides:
\[ \tan x = \sqrt{3} \quad \text{or} \quad \tan x = -\sqrt{3} \]

### Step 4: Find the angles corresponding to \( \tan x = \sqrt{3} \)
The value \( \tan x = \sqrt{3} \) corresponds to:
\[ x = 60^\circ + n \cdot 180^\circ \quad \text{for any integer } n \]

### Step 5: Find the angles corresponding to \( \tan x = -\sqrt{3} \)
The value \( \tan x = -\sqrt{3} \) corresponds to:
\[ x = 120^\circ + n \cdot 180^\circ \quad \text{for any integer } n \]

### Conclusion
For the principal value in the range \( 0^\circ \) to \( 360^\circ \), the solutions are:
- From \( \tan x = \sqrt{3} \): \( x = 60^\circ \)
- From \( \tan x = -\sqrt{3} \): \( x = 120^\circ \)

Since the problem asks for \( x \) in the context of the options provided, we can conclude that:
\[ x = 60^\circ \]