` write tan 8 alpha ` in temrs of ` tan 4 alpha .`

Answer

To express \( \tan 8\alpha \) in terms of \( \tan 4\alpha \), we can use the double angle formula for tangent. The formula states:

\[ \tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta} \]

### Step-by-Step Solution:

1. **Identify the angle**: We want to express \( \tan 8\alpha \), which can be seen as \( \tan 2(4\alpha) \). Here, we can let \( \theta = 4\alpha \).

2. **Apply the double angle formula**: Substitute \( \theta \) with \( 4\alpha \) in the double angle formula:
\[ \tan 8\alpha = \tan 2(4\alpha) = \frac{2\tan(4\alpha)}{1 - \tan^2(4\alpha)} \]

3. **Write the final expression**: Thus, we have:
\[ \tan 8\alpha = \frac{2\tan(4\alpha)}{1 - \tan^2(4\alpha)} \]

### Final Result:
\[ \tan 8\alpha = \frac{2\tan(4\alpha)}{1 - \tan^2(4\alpha)} \]

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