If `xsin60^(@). tan30^(@)=sec60^(@).cot45^(@)` , then the value of x is

Options

A. 2
B. `2sqrt(3)`
C. 4
D. `4sqrt(3)`

Answer

To solve the equation \( x \sin 60^\circ \tan 30^\circ = \sec 60^\circ \cot 45^\circ \), we will first substitute the known values of the trigonometric functions involved.

### Step 1: Substitute the values of trigonometric functions
- \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)
- \( \tan 30^\circ = \frac{1}{\sqrt{3}} \)
- \( \sec 60^\circ = 2 \)
- \( \cot 45^\circ = 1 \)

Now, substituting these values into the equation:

\[ x \left( \frac{\sqrt{3}}{2} \right) \left( \frac{1}{\sqrt{3}} \right) = 2 \cdot 1 \]

### Step 2: Simplify the left-hand side
The left-hand side simplifies as follows:

\[ x \left( \frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{3}} \right) = x \left( \frac{1}{2} \right) \]

So the equation now looks like:

\[ \frac{x}{2} = 2 \]

### Step 3: Solve for \( x \)
To isolate \( x \), multiply both sides by 2:

\[ x = 2 \cdot 2 \]

Thus,

\[ x = 4 \]

### Final Answer
The value of \( x \) is \( 4 \).

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