If ` x tan 60^(@) xx tan^(3) 30^(@) xx tan ^(2) 45^(@) = 1 ` ,then `x =` ?

Options

A. `3`
B. ` sqrt""3 `
C. `1`
D. None of these

Answer

To solve the equation \( x \tan 60^\circ \times \tan^3 30^\circ \times \tan^2 45^\circ = 1 \), we first need to determine the values of the tangent functions involved.

### Step 1: Find the values of the tangent functions
- \( \tan 60^\circ = \sqrt{3} \)
- \( \tan 30^\circ = \frac{1}{\sqrt{3}} \)
- \( \tan 45^\circ = 1 \)

### Step 2: Substitute the values into the equation
Substituting the values into the equation gives us:
\[ x \times \sqrt{3} \times \left(\frac{1}{\sqrt{3}}\right)^3 \times 1^2 = 1 \]

### Step 3: Simplify the equation
Now we simplify the equation:
\[ x \times \sqrt{3} \times \left(\frac{1}{3\sqrt{3}}\right) = 1 \]
This simplifies to:
\[ x \times \frac{\sqrt{3}}{3\sqrt{3}} = 1 \]
\[ x \times \frac{1}{3} = 1 \]

### Step 4: Solve for \( x \)
To isolate \( x \), multiply both sides by 3:
\[ x = 3 \]

Thus, the value of \( x \) is \( 3 \).

### Final Answer:
\[ x = 3 \]

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