`(kcosec^(2)30^(@).sec^(2)45)/(8cos^(2)45^(@).sin^(2)60^(@))=tan^(2)60^(@)-tan^(2)30^(@)`, then k = ?

Options

A. 1
B. `-1`
C. 2
D. 0

Answer

To solve the equation

\[ \frac{k \cdot \cos^2(30^\circ) \cdot \sec^2(45^\circ)}{8 \cdot \cos^2(45^\circ) \cdot \sin^2(60^\circ)} = \tan^2(60^\circ) - \tan^2(30^\circ) \]

we will start by substituting the known values of the trigonometric functions.

### Step 1: Substitute the known values

1. **Values of trigonometric functions**:
- \(\cos(30^\circ) = \frac{\sqrt{3}}{2} \Rightarrow \cos^2(30^\circ) = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}\)
- \(\sec(45^\circ) = \sqrt{2} \Rightarrow \sec^2(45^\circ) = (\sqrt{2})^2 = 2\)
- \(\cos(45^\circ) = \frac{1}{\sqrt{2}} \Rightarrow \cos^2(45^\circ) = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}\)
- \(\sin(60^\circ) = \frac{\sqrt{3}}{2} \Rightarrow \sin^2(60^\circ) = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}\)
- \(\tan(60^\circ) = \sqrt{3} \Rightarrow \tan^2(60^\circ) = (\sqrt{3})^2 = 3\)
- \(\tan(30^\circ) = \frac{1}{\sqrt{3}} \Rightarrow \tan^2(30^\circ) = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3}\)

Now substituting these values into the equation:

\[ \frac{k \cdot \frac{3}{4} \cdot 2}{8 \cdot \frac{1}{2} \cdot \frac{3}{4}} = 3 - \frac{1}{3} \]

### Step 2: Simplify the left side

2. **Simplifying the left side**:

\[ \frac{k \cdot \frac{3}{4} \cdot 2}{8 \cdot \frac{1}{2} \cdot \frac{3}{4}} = \frac{k \cdot \frac{3}{2}}{8 \cdot \frac{1}{2} \cdot \frac{3}{4}} \]

Calculating the denominator:

\[ 8 \cdot \frac{1}{2} \cdot \frac{3}{4} = 8 \cdot \frac{3}{8} = 3 \]

So the left side becomes:

\[ \frac{k \cdot \frac{3}{2}}{3} = \frac{k}{2} \]

### Step 3: Simplify the right side

3. **Simplifying the right side**:

Calculating \(3 - \frac{1}{3}\):

\[ 3 - \frac{1}{3} = \frac{9}{3} - \frac{1}{3} = \frac{8}{3} \]

### Step 4: Set the simplified expressions equal

4. **Setting the two sides equal**:

Now we have:

\[ \frac{k}{2} = \frac{8}{3} \]

### Step 5: Solve for \(k\)

5. **Solving for \(k\)**:

Multiply both sides by 2:

\[ k = \frac{8}{3} \cdot 2 = \frac{16}{3} \]

### Final Answer

Thus, the value of \(k\) is:

\[ k = \frac{16}{3} \]