If `tan theta+cot theta=4`, then find the value of `tan^(4)theta+cot^(4)theta`.

Answer

To solve the problem, we need to find the value of \( \tan^4 \theta + \cot^4 \theta \) given that \( \tan \theta + \cot \theta = 4 \).

### Step-by-step Solution:

1. **Start with the given equation:**
\[ \tan \theta + \cot \theta = 4 \]

2. **Square both sides to find \( \tan^2 \theta + \cot^2 \theta \):**
\[ (\tan \theta + \cot \theta)^2 = 4^2 \]
Expanding the left side using the identity \( (a + b)^2 = a^2 + b^2 + 2ab \):
\[ \tan^2 \theta + \cot^2 \theta + 2 \tan \theta \cot \theta = 16 \]

3. **Use the identity \( \tan \theta \cot \theta = 1 \):**
\[ 2 \tan \theta \cot \theta = 2 \cdot 1 = 2 \]
Substitute this into the equation:
\[ \tan^2 \theta + \cot^2 \theta + 2 = 16 \]
Simplifying gives:
\[ \tan^2 \theta + \cot^2 \theta = 16 - 2 = 14 \]

4. **Now square \( \tan^2 \theta + \cot^2 \theta \) to find \( \tan^4 \theta + \cot^4 \theta \):**
\[ (\tan^2 \theta + \cot^2 \theta)^2 = 14^2 \]
Expanding this gives:
\[ \tan^4 \theta + \cot^4 \theta + 2 \tan^2 \theta \cot^2 \theta = 196 \]

5. **Substituting \( \tan^2 \theta \cot^2 \theta = 1 \):**
\[ 2 \tan^2 \theta \cot^2 \theta = 2 \cdot 1 = 2 \]
Substitute this into the equation:
\[ \tan^4 \theta + \cot^4 \theta + 2 = 196 \]
Simplifying gives:
\[ \tan^4 \theta + \cot^4 \theta = 196 - 2 = 194 \]

### Final Answer:
\[ \tan^4 \theta + \cot^4 \theta = 194 \]