if `x sin ^2 60^@ -3/2 sec 60^@ tan ^2 30^@ + 4/5 sin ^2 45^@ tan^2 60^@` = 0 then x is

Options

A. `-(1)/(15)`
B. `-4`
C. `-(4)/(15)`
D. `-2`

Answer

To solve the equation

\[ x \sin^2 60^\circ - \frac{3}{2} \sec 60^\circ \tan^2 30^\circ + \frac{4}{5} \sin^2 45^\circ \tan^2 60^\circ = 0, \]

we will substitute the trigonometric values and simplify step by step.

### Step 1: Substitute the trigonometric values

We know the following values:
- \(\sin 60^\circ = \frac{\sqrt{3}}{2}\)
- \(\sec 60^\circ = 2\)
- \(\tan 30^\circ = \frac{1}{\sqrt{3}}\)
- \(\sin 45^\circ = \frac{1}{\sqrt{2}}\)
- \(\tan 60^\circ = \sqrt{3}\)

Substituting these values into the equation gives:

\[ x \left(\frac{\sqrt{3}}{2}\right)^2 - \frac{3}{2} \cdot 2 \cdot \left(\frac{1}{\sqrt{3}}\right)^2 + \frac{4}{5} \cdot \left(\frac{1}{\sqrt{2}}\right)^2 \cdot (\sqrt{3})^2 = 0. \]

### Step 2: Simplify each term

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:

\[ x \cdot \frac{3}{4} - \frac{3}{2} \cdot 2 \cdot \frac{1}{3} + \frac{4}{5} \cdot \frac{1}{2} \cdot 3 = 0. \]

### Step 3: Simplify further

Now we simplify the equation:

\[ x \cdot \frac{3}{4} - 1 + \frac{6}{5} = 0. \]

### Step 4: Combine like terms

To combine the constants, we need a common denominator. The common denominator of 1 and 5 is 5:

\[ x \cdot \frac{3}{4} - 1 + \frac{6}{5} = x \cdot \frac{3}{4} - \frac{5}{5} + \frac{6}{5} = x \cdot \frac{3}{4} + \frac{1}{5} = 0. \]

### Step 5: Isolate \(x\)

Now we isolate \(x\):

\[ x \cdot \frac{3}{4} = -\frac{1}{5}. \]

### Step 6: Solve for \(x\)

Multiply both sides by \(\frac{4}{3}\):

\[ x = -\frac{1}{5} \cdot \frac{4}{3} = -\frac{4}{15}. \]

Thus, the value of \(x\) is

\[ \boxed{-\frac{4}{15}}. \]