Find x if x `sin^2 60^@ -1/2 "sec"60^@ tan^2 30^@ + 4/3 "sin"^2 45^@ tan^2 60^@` = 0

Answer

To solve the equation

\[ x \sin^2 60^\circ - \frac{1}{2} \sec 60^\circ \tan^2 30^\circ + \frac{4}{3} \sin^2 45^\circ \tan^2 60^\circ = 0, \]

we will follow these steps:

### Step 1: Calculate the trigonometric values
First, we need to find the values of the trigonometric functions involved in the equation.

- \(\sin 60^\circ = \frac{\sqrt{3}}{2}\)
- \(\sec 60^\circ = \frac{1}{\cos 60^\circ} = \frac{1}{\frac{1}{2}} = 2\)
- \(\tan 30^\circ = \frac{1}{\sqrt{3}}\)
- \(\sin 45^\circ = \frac{1}{\sqrt{2}}\)
- \(\tan 60^\circ = \sqrt{3}\)

### Step 2: Substitute the values into the equation
Now, substitute these values into the equation:

\[ x \left(\frac{\sqrt{3}}{2}\right)^2 - \frac{1}{2} \cdot 2 \cdot \left(\frac{1}{\sqrt{3}}\right)^2 + \frac{4}{3} \cdot \left(\frac{1}{\sqrt{2}}\right)^2 \cdot (\sqrt{3})^2 = 0. \]

### Step 3: Simplify the equation
Calculating each term:

1. \(\left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}\)
2. \(\left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3}\)
3. \(\left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}\)
4. \((\sqrt{3})^2 = 3\)

Now, substituting these values back into the equation gives:

\[ x \cdot \frac{3}{4} - \frac{1}{2} \cdot 2 \cdot \frac{1}{3} + \frac{4}{3} \cdot \frac{1}{2} \cdot 3 = 0. \]

This simplifies to:

\[ x \cdot \frac{3}{4} - \frac{1}{3} + 2 = 0. \]

### Step 4: Combine like terms
Now, we can combine the constant terms:

\[ x \cdot \frac{3}{4} + \left(2 - \frac{1}{3}\right) = 0. \]

To combine \(2\) and \(-\frac{1}{3}\), convert \(2\) into a fraction:

\[ 2 = \frac{6}{3} \implies 2 - \frac{1}{3} = \frac{6}{3} - \frac{1}{3} = \frac{5}{3}. \]

So the equation now looks like:

\[ x \cdot \frac{3}{4} + \frac{5}{3} = 0. \]

### Step 5: Solve for \(x\)
Now, isolate \(x\):

\[ x \cdot \frac{3}{4} = -\frac{5}{3}. \]

Now, multiply both sides by \(\frac{4}{3}\):

\[ x = -\frac{5}{3} \cdot \frac{4}{3} = -\frac{20}{9}. \]

### Final Answer
Thus, the value of \(x\) is

\[ \boxed{-\frac{20}{9}}. \]