If `sin^4x/2+cos^4x/3=1/5`then

Options

A. `tan^2x=2/3`
B. `sin^8x/8+cos^8x/27=1/125`
C. `tan^2x=1/3`
D. `sin^8/8+cos^8x/27=2/125`

Explanation

To solve the equation \( \frac{\sin^4 x}{2} + \frac{\cos^4 x}{3} = \frac{1}{5} \), we will follow these steps:

### Step 1: Rewrite \( \cos^4 x \)
Using the Pythagorean identity \( \cos^2 x = 1 - \sin^2 x \), we can express \( \cos^4 x \) as:
\[ \cos^4 x = (1 - \sin^2 x)^2 \]

### Step 2: Substitute in the equation
Substituting \( \cos^4 x \) into the original equation gives:
\[ \frac{\sin^4 x}{2} + \frac{(1 - \sin^2 x)^2}{3} = \frac{1}{5} \]

### Step 3: Clear the fractions
To eliminate the denominators, we can multiply the entire equation by 30 (the least common multiple of 2, 3, and 5):
\[ 30 \left( \frac{\sin^4 x}{2} \right) + 30 \left( \frac{(1 - \sin^2 x)^2}{3} \right) = 30 \left( \frac{1}{5} \right) \]
This simplifies to:
\[ 15 \sin^4 x + 10 (1 - \sin^2 x)^2 = 6 \]

### Step 4: Expand the equation
Now, expand \( (1 - \sin^2 x)^2 \):
\[ (1 - \sin^2 x)^2 = 1 - 2\sin^2 x + \sin^4 x \]
Substituting this back into the equation gives:
\[ 15 \sin^4 x + 10(1 - 2\sin^2 x + \sin^4 x) = 6 \]
This simplifies to:
\[ 15 \sin^4 x + 10 - 20 \sin^2 x + 10 \sin^4 x = 6 \]

### Step 5: Combine like terms
Combine the terms involving \( \sin^4 x \):
\[ (15 \sin^4 x + 10 \sin^4 x) - 20 \sin^2 x + 10 = 6 \]
This simplifies to:
\[ 25 \sin^4 x - 20 \sin^2 x + 10 - 6 = 0 \]
Thus, we have:
\[ 25 \sin^4 x - 20 \sin^2 x + 4 = 0 \]

### Step 6: Substitute \( t = \sin^2 x \)
Let \( t = \sin^2 x \). Then \( \sin^4 x = t^2 \), and we can rewrite the equation as:
\[ 25 t^2 - 20 t + 4 = 0 \]

### Step 7: Solve the quadratic equation
Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ t = \frac{20 \pm \sqrt{(-20)^2 - 4 \cdot 25 \cdot 4}}{2 \cdot 25} \]
Calculating the discriminant:
\[ t = \frac{20 \pm \sqrt{400 - 400}}{50} = \frac{20 \pm 0}{50} = \frac{20}{50} = \frac{2}{5} \]

### Step 8: Find \( \sin^2 x \)
Since \( t = \sin^2 x \), we have:
\[ \sin^2 x = \frac{2}{5} \]

### Step 9: Find \( \cos^2 x \)
Using the identity \( \cos^2 x = 1 - \sin^2 x \):
\[ \cos^2 x = 1 - \frac{2}{5} = \frac{3}{5} \]

### Step 10: Find \( \tan^2 x \)
Now, we can find \( \tan^2 x \):
\[ \tan^2 x = \frac{\sin^2 x}{\cos^2 x} = \frac{\frac{2}{5}}{\frac{3}{5}} = \frac{2}{3} \]

### Conclusion
The value of \( \tan^2 x \) is \( \frac{2}{3} \).