If a `sin45^(@).cos45^(@).tan60^(@)=tan^(2)45^(@)-cos60^(@)`, then find the value of a .

Options

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

Answer

To solve the equation \( a \sin 45^\circ \cos 45^\circ \tan 60^\circ = \tan^2 45^\circ - \cos^2 60^\circ \), we will follow these steps:

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

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

### Step 2: Simplify the left side of the equation
The left side simplifies as follows:
\[ a \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} \cdot \sqrt{3}\right) = a \left(\frac{1}{2} \cdot \sqrt{3}\right) = \frac{a \sqrt{3}}{2} \]

### Step 3: Simplify the right side of the equation
The right side simplifies as follows:
\[ 1 - \frac{1}{2} = \frac{1}{2} \]

### Step 4: Set the left side equal to the right side
Now we have:
\[ \frac{a \sqrt{3}}{2} = \frac{1}{2} \]

### Step 5: Solve for \( a \)
To isolate \( a \), we multiply both sides by 2:
\[ a \sqrt{3} = 1 \]
Now, divide both sides by \( \sqrt{3} \):
\[ a = \frac{1}{\sqrt{3}} \]

### Final Answer
Thus, the value of \( a \) is:
\[ \boxed{\frac{1}{\sqrt{3}}} \]