If `(2tan^(2)30^(@))/(1-tan^(2)30^(@))+sec^(2)45^(@)-sec^(2)0^(@)=xsec60^(@)` , then the value of x is

Options

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

Answer

To solve the equation

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

we will follow these steps:

### Step 1: Calculate \(\tan 30^\circ\)

We know that:

\[ \tan 30^\circ = \frac{1}{\sqrt{3}}. \]

### Step 2: Calculate \(\tan^2 30^\circ\)

Now, we find:

\[ \tan^2 30^\circ = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3}. \]

### Step 3: Substitute \(\tan^2 30^\circ\) into the equation

Now substitute \(\tan^2 30^\circ\) into the equation:

\[ \frac{2 \cdot \frac{1}{3}}{1 - \frac{1}{3}}. \]

### Step 4: Simplify the denominator

Calculate the denominator:

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

### Step 5: Substitute and simplify

Now substitute back into the equation:

\[ \frac{2 \cdot \frac{1}{3}}{\frac{2}{3}} = \frac{2/3}{2/3} = 1. \]

### Step 6: Calculate \(\sec^2 45^\circ\) and \(\sec^2 0^\circ\)

We know that:

\[ \sec 45^\circ = \sqrt{2} \implies \sec^2 45^\circ = 2, \]
\[ \sec 0^\circ = 1 \implies \sec^2 0^\circ = 1. \]

### Step 7: Substitute \(\sec^2 45^\circ\) and \(\sec^2 0^\circ\) into the equation

Now substitute these values into the equation:

\[ 1 + 2 - 1 = x \sec 60^\circ. \]

### Step 8: Calculate \(\sec 60^\circ\)

We know that:

\[ \sec 60^\circ = 2. \]

### Step 9: Substitute \(\sec 60^\circ\) into the equation

Now substitute back:

\[ 1 + 2 - 1 = x \cdot 2. \]

### Step 10: Simplify the left side

This simplifies to:

\[ 2 = 2x. \]

### Step 11: Solve for \(x\)

Dividing both sides by 2 gives:

\[ x = 1. \]

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

\[ \boxed{1}. \]