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 the equation
We start with the given equation:
\[ \frac{\sin^4 x}{2} + \frac{\cos^4 x}{3} = \frac{1}{5} \]

### Step 2: Find a common denominator
To combine the fractions on the left-hand side, we find a common denominator, which is 6:
\[ \frac{3\sin^4 x}{6} + \frac{2\cos^4 x}{6} = \frac{1}{5} \]
This simplifies to:
\[ 3\sin^4 x + 2\cos^4 x = \frac{6}{5} \]

### Step 3: Substitute \(\cos^4 x\)
Using the identity \(\cos^2 x = 1 - \sin^2 x\), we can express \(\cos^4 x\) as:
\[ \cos^4 x = (1 - \sin^2 x)^2 \]
Substituting this into the equation gives:
\[ 3\sin^4 x + 2(1 - \sin^2 x)^2 = \frac{6}{5} \]

### Step 4: Expand the equation
Expanding \(2(1 - \sin^2 x)^2\):
\[ 2(1 - 2\sin^2 x + \sin^4 x) = 2 - 4\sin^2 x + 2\sin^4 x \]
Now, substituting this back into the equation:
\[ 3\sin^4 x + 2 - 4\sin^2 x + 2\sin^4 x = \frac{6}{5} \]
Combining like terms:
\[ 5\sin^4 x - 4\sin^2 x + 2 = \frac{6}{5} \]

### Step 5: Clear the fraction
Multiply through by 5 to eliminate the fraction:
\[ 25\sin^4 x - 20\sin^2 x + 10 = 6 \]
This simplifies to:
\[ 25\sin^4 x - 20\sin^2 x + 4 = 0 \]

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

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

### Step 8: Find \(\cos^2 x\)
Since \(y = \sin^2 x\):
\[ \sin^2 x = \frac{2}{5} \]
Then, using the Pythagorean identity:
\[ \cos^2 x = 1 - \sin^2 x = 1 - \frac{2}{5} = \frac{3}{5} \]

### Step 9: 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
Thus, the solution to the original equation is:
\[ \tan^2 x = \frac{2}{3} \]