The value of x which stasfies the equation ` 2 cosec^2 30^@ + x sin^2 60 ^@ - 3/4 tan ^2 30^@ = 10`

Options

A. 2
B. 3
C. 0
D. 1

Answer

To solve the equation \( 2 \csc^2 30^\circ + x \sin^2 60^\circ - \frac{3}{4} \tan^2 30^\circ = 10 \), we will follow these steps:

### Step 1: Calculate \( \csc^2 30^\circ \)
The cosecant function is the reciprocal of the sine function. We know that:
\[ \sin 30^\circ = \frac{1}{2} \]
Thus,
\[ \csc 30^\circ = \frac{1}{\sin 30^\circ} = \frac{1}{\frac{1}{2}} = 2 \]
Therefore,
\[ \csc^2 30^\circ = 2^2 = 4 \]

### Step 2: Substitute \( \csc^2 30^\circ \) into the equation
Now substituting \( \csc^2 30^\circ \) into the equation:
\[ 2 \cdot 4 + x \sin^2 60^\circ - \frac{3}{4} \tan^2 30^\circ = 10 \]
This simplifies to:
\[ 8 + x \sin^2 60^\circ - \frac{3}{4} \tan^2 30^\circ = 10 \]

### Step 3: Calculate \( \sin^2 60^\circ \)
We know that:
\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \]
Thus,
\[ \sin^2 60^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

### Step 4: Calculate \( \tan^2 30^\circ \)
We know that:
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \]
Thus,
\[ \tan^2 30^\circ = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3} \]

### Step 5: Substitute \( \sin^2 60^\circ \) and \( \tan^2 30^\circ \) into the equation
Substituting these values into the equation gives:
\[ 8 + x \cdot \frac{3}{4} - \frac{3}{4} \cdot \frac{1}{3} = 10 \]
This simplifies to:
\[ 8 + \frac{3x}{4} - \frac{1}{4} = 10 \]

### Step 6: Simplify the equation
Combining the constants:
\[ 8 - \frac{1}{4} = \frac{32}{4} - \frac{1}{4} = \frac{31}{4} \]
So the equation now is:
\[ \frac{31}{4} + \frac{3x}{4} = 10 \]

### Step 7: Eliminate the fraction
Multiply the entire equation by 4 to eliminate the fraction:
\[ 31 + 3x = 40 \]

### Step 8: Solve for \( x \)
Now, isolate \( x \):
\[ 3x = 40 - 31 \]
\[ 3x = 9 \]
\[ x = 3 \]

### Final Answer
Thus, the value of \( x \) that satisfies the equation is:
\[ \boxed{3} \]