If `(sin^(2)x-2cos^(2)x+1)/(sin^(2)x+2cos^(2)x-1)=4`, then the value of `2 tan^(2)x` is

Options

A. 3
B. 4
C. 5
D. 6

Explanation

To solve the equation

\[ \frac{\sin^2 x - 2\cos^2 x + 1}{\sin^2 x + 2\cos^2 x - 1} = 4, \]

we will follow these steps:

### Step 1: Cross-Multiply
We start by cross-multiplying to eliminate the fraction:

\[ \sin^2 x - 2\cos^2 x + 1 = 4(\sin^2 x + 2\cos^2 x - 1). \]

### Step 2: Expand the Right Side
Expanding the right side gives us:

\[ \sin^2 x - 2\cos^2 x + 1 = 4\sin^2 x + 8\cos^2 x - 4. \]

### Step 3: Rearranging the Equation
Now, we will rearrange the equation by moving all terms to one side:

\[ \sin^2 x - 2\cos^2 x + 1 - 4\sin^2 x - 8\cos^2 x + 4 = 0. \]

Combining like terms:

\[ -3\sin^2 x - 10\cos^2 x + 5 = 0. \]

### Step 4: Rearranging Further
Rearranging gives:

\[ 3\sin^2 x + 10\cos^2 x - 5 = 0. \]

### Step 5: Divide by \(\cos^2 x\)
Now, we divide the entire equation by \(\cos^2 x\):

\[ 3\frac{\sin^2 x}{\cos^2 x} + 10 - 5\frac{1}{\cos^2 x} = 0. \]

This simplifies to:

\[ 3\tan^2 x + 10 - 5\sec^2 x = 0. \]

### Step 6: Substitute \(\sec^2 x\)
Recall that \(\sec^2 x = 1 + \tan^2 x\). Substitute this into the equation:

\[ 3\tan^2 x + 10 - 5(1 + \tan^2 x) = 0. \]

### Step 7: Simplify
Expanding gives:

\[ 3\tan^2 x + 10 - 5 - 5\tan^2 x = 0. \]

Combining like terms:

\[ -2\tan^2 x + 5 = 0. \]

### Step 8: Solve for \(\tan^2 x\)
Rearranging gives:

\[ 2\tan^2 x = 5. \]

### Step 9: Conclusion
Thus, the value of \(2\tan^2 x\) is:

\[ \boxed{5}. \]