If `2x^(2)cos 60^(@)-4cot^(2)45^(@)-2tan 60^(@)=0`, then what is the value of `x` ?

Options

A. `2`
B. `3`
C. `sqrt(3)-1`
D. `sqrt(3)+1`

Answer

To solve the equation \( 2x^2 \cos 60^\circ - 4 \cot^2 45^\circ - 2 \tan 60^\circ = 0 \), we will follow these steps:

### Step 1: Substitute the values of trigonometric functions
We know:
- \( \cos 60^\circ = \frac{1}{2} \)
- \( \cot 45^\circ = 1 \) (since \( \cot 45^\circ = \frac{1}{\tan 45^\circ} = 1 \))
- \( \tan 60^\circ = \sqrt{3} \)

Substituting these values into the equation gives:
\[ 2x^2 \left(\frac{1}{2}\right) - 4(1)^2 - 2(\sqrt{3}) = 0 \]

### Step 2: Simplify the equation
Now simplify the equation:
\[ x^2 - 4 - 2\sqrt{3} = 0 \]

### Step 3: Rearrange the equation
Rearranging the equation gives:
\[ x^2 = 4 + 2\sqrt{3} \]

### Step 4: Recognize the form of the equation
Notice that \( 4 + 2\sqrt{3} \) can be expressed as a perfect square:
\[ 4 + 2\sqrt{3} = (\sqrt{3} + 1)^2 \]

### Step 5: Take the square root
Taking the square root of both sides:
\[ x = \sqrt{(\sqrt{3} + 1)^2} \]
Thus,
\[ x = \sqrt{3} + 1 \]

### Conclusion
The value of \( x \) is \( \sqrt{3} + 1 \).