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

Options

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

Answer

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: Substitute \( \sin^4 x \) and \( \cos^4 x \)
Recall that \( \sin^4 x = (\sin^2 x)^2 \) and \( \cos^4 x = (\cos^2 x)^2 \). We can express the equation as:
\[ \frac{(\sin^2 x)^2}{2} + \frac{(\cos^2 x)^2}{3} = \frac{1}{5} \]

### Step 3: Let \( \sin^2 x = a \) and \( \cos^2 x = b \)
Using the identity \( a + b = 1 \) (since \( \sin^2 x + \cos^2 x = 1 \)), we can rewrite \( b \) as \( 1 - a \). Thus, we have:
\[ \frac{a^2}{2} + \frac{(1-a)^2}{3} = \frac{1}{5} \]

### Step 4: Expand \( (1-a)^2 \)
Now, expand \( (1-a)^2 \):
\[ (1-a)^2 = 1 - 2a + a^2 \]
Substituting this back into the equation gives:
\[ \frac{a^2}{2} + \frac{1 - 2a + a^2}{3} = \frac{1}{5} \]

### Step 5: Combine the fractions
To combine the fractions, we need a common denominator, which is 6:
\[ \frac{3a^2}{6} + \frac{2(1 - 2a + a^2)}{6} = \frac{1}{5} \]
This simplifies to:
\[ \frac{3a^2 + 2 - 4a + 2a^2}{6} = \frac{1}{5} \]
Combining like terms gives:
\[ \frac{5a^2 - 4a + 2}{6} = \frac{1}{5} \]

### Step 6: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 5(5a^2 - 4a + 2) = 6 \]
Expanding this results in:
\[ 25a^2 - 20a + 10 = 6 \]

### Step 7: Rearranging the equation
Rearranging gives:
\[ 25a^2 - 20a + 4 = 0 \]

### Step 8: Solve the quadratic equation
Using the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( A = 25, B = -20, C = 4 \):
\[ D = (-20)^2 - 4 \cdot 25 \cdot 4 = 400 - 400 = 0 \]
Since the discriminant \( D = 0 \), there is one real solution:
\[ a = \frac{20}{2 \cdot 25} = \frac{20}{50} = \frac{2}{5} \]

### Step 9: Find \( \sin^2 x \) and \( \cos^2 x \)
Thus, we have:
\[ \sin^2 x = a = \frac{2}{5} \]
Using \( \cos^2 x = 1 - \sin^2 x \):
\[ \cos^2 x = 1 - \frac{2}{5} = \frac{3}{5} \]

### Step 10: Find \( \tan^2 x \)
Using the identity \( \tan^2 x = \frac{\sin^2 x}{\cos^2 x} \):
\[ \tan^2 x = \frac{\frac{2}{5}}{\frac{3}{5}} = \frac{2}{3} \]

### Final Result
Thus, we have:
\[ \sin^2 x = \frac{2}{5}, \quad \cos^2 x = \frac{3}{5}, \quad \tan^2 x = \frac{2}{3} \]