If `xtan45^(@).cos60^(@)=sin60^(@)cot60^(@)`, then x is equal to

Options

A. 1
B. `1//2`
C. `sqrt3`
D. `1//sqrt2`

Answer

To solve the equation \( x \tan 45^\circ \cdot \cos 60^\circ = \sin 60^\circ \cdot \cot 60^\circ \), we will follow these steps:

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

Substituting these values into the equation gives us:
\[ x \cdot 1 \cdot \frac{1}{2} = \frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{3}} \]

### Step 2: Simplify the right-hand side
Now simplify the right-hand side:
\[ \frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{3}} = \frac{1}{2} \]

### Step 3: Set up the equation
Now we have:
\[ x \cdot \frac{1}{2} = \frac{1}{2} \]

### Step 4: Solve for \( x \)
To isolate \( x \), multiply both sides by 2:
\[ x = 1 \]

### Conclusion
Thus, the value of \( x \) is \( 1 \).

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