Solve `sin^(4)x=1+tan^(8)x`.

Explanation

To solve the equation \( \sin^4 x = 1 + \tan^8 x \), we will follow these steps:

### Step 1: Rewrite the equation using trigonometric identities
We know that \( \tan x = \frac{\sin x}{\cos x} \). Therefore, we can express \( \tan^8 x \) as:
\[ \tan^8 x = \left( \frac{\sin^2 x}{\cos^2 x} \right)^4 = \frac{\sin^8 x}{\cos^8 x} \]
Substituting this into the equation gives us:
\[ \sin^4 x = 1 + \frac{\sin^8 x}{\cos^8 x} \]

### Step 2: Multiply through by \( \cos^8 x \)
To eliminate the fraction, we multiply both sides by \( \cos^8 x \):
\[ \sin^4 x \cos^8 x = \cos^8 x + \sin^8 x \]

### Step 3: Rearranging the equation
Rearranging the equation gives us:
\[ \sin^4 x \cos^8 x - \sin^8 x - \cos^8 x = 0 \]

### Step 4: Factor the equation
This equation can be factored. Notice that we can group the terms:
\[ \sin^4 x (\cos^8 x - \sin^4 x) - \cos^8 x = 0 \]

### Step 5: Analyze the factors
For the equation to hold true, either:
1. \( \sin^4 x = 0 \) or
2. \( \cos^8 x - \sin^4 x - \cos^8 x = 0 \)

### Step 6: Solve \( \sin^4 x = 0 \)
From \( \sin^4 x = 0 \), we find:
\[ \sin x = 0 \]
This occurs at:
\[ x = n\pi, \quad n \in \mathbb{Z} \]

### Step 7: Check the second factor
Now we check the second factor:
\[ \cos^8 x - \sin^4 x - \cos^8 x = 0 \implies -\sin^4 x = 0 \]
This leads us back to \( \sin^4 x = 0 \), which we already solved.

### Conclusion
Thus, the only solutions to the equation \( \sin^4 x = 1 + \tan^8 x \) are:
\[ x = n\pi, \quad n \in \mathbb{Z} \]