The general solution of `4 sin^4 x + cos^4x= 1` is

Options

A. `n pi pm alpha//2, alpha=cos^(-1) (1//5), AA n in Z`
B. `n pi pm alpha//2, alpha=cos^(-1) (3//5), AA n in Z`
C. `2 n pi pm alpha//2, alpha=cos^(-1) (1//3), AA n in Z`
D. none of these

Explanation

To find the general solution of the equation \(4 \sin^4 x + \cos^4 x = 1\), we can follow these steps:

### Step-by-Step Solution:

1. **Rewrite the equation**:
\[ 4 \sin^4 x + \cos^4 x = 1 \]

2. **Use the identity for \(\sin^4 x\)**:
Recall that \(\sin^4 x = (\sin^2 x)^2\) and \(\cos^4 x = (\cos^2 x)^2\). Let \(y = \sin^2 x\). Then, \(\cos^2 x = 1 - y\).
\[ 4y^2 + (1 - y)^2 = 1 \]

3. **Expand the equation**:
\[ 4y^2 + (1 - 2y + y^2) = 1 \]
\[ 4y^2 + 1 - 2y + y^2 = 1 \]

4. **Combine like terms**:
\[ 5y^2 - 2y + 1 - 1 = 0 \]
\[ 5y^2 - 2y = 0 \]

5. **Factor the equation**:
\[ y(5y - 2) = 0 \]

6. **Find the roots**:
- \(y = 0\) or \(5y - 2 = 0 \Rightarrow y = \frac{2}{5}\)

7. **Recall that \(y = \sin^2 x\)**:
- For \(y = 0\):
\[ \sin^2 x = 0 \Rightarrow \sin x = 0 \Rightarrow x = n\pi, \quad n \in \mathbb{Z} \]
- For \(y = \frac{2}{5}\):
\[ \sin^2 x = \frac{2}{5} \Rightarrow \sin x = \pm \sqrt{\frac{2}{5}} \]

8. **Find the angles for \(\sin x = \sqrt{\frac{2}{5}}\)** and \(\sin x = -\sqrt{\frac{2}{5}}\)**:
- For \(\sin x = \sqrt{\frac{2}{5}}\):
\[ x = \frac{\pi}{2} - \theta + 2n\pi \quad \text{and} \quad x = \frac{\pi}{2} + \theta + 2n\pi \]
- For \(\sin x = -\sqrt{\frac{2}{5}}\):
\[ x = -\frac{\pi}{2} - \theta + 2n\pi \quad \text{and} \quad x = -\frac{\pi}{2} + \theta + 2n\pi \]

9. **Combine the solutions**:
The general solution can be expressed as:
\[ x = n\pi \quad \text{or} \quad x = \arcsin\left(\sqrt{\frac{2}{5}}\right) + 2n\pi \quad \text{or} \quad x = \pi - \arcsin\left(\sqrt{\frac{2}{5}}\right) + 2n\pi \]
\[ x = n\pi \quad \text{or} \quad x = -\arcsin\left(\sqrt{\frac{2}{5}}\right) + 2n\pi \quad \text{or} \quad x = -\pi + \arcsin\left(\sqrt{\frac{2}{5}}\right) + 2n\pi \]

### Final General Solution:
\[ x = n\pi \quad \text{or} \quad x = n\pi + (-1)^n \arcsin\left(\sqrt{\frac{2}{5}}\right) \]