If `tan^(5)thetatan^(5)5theta=1`, then find the value of `tan^(4)3theta`.

Options

A. 5
B. `-1`
C. 1
D. 3

Answer

To solve the equation \( \tan^5 \theta \tan^5 5\theta = 1 \) and find the value of \( \tan^4 3\theta \), we can follow these steps:

### Step 1: Rewrite the given equation
We start with the equation:
\[ \tan^5 \theta \tan^5 5\theta = 1 \]
This implies:
\[ \tan^5 \theta = \frac{1}{\tan^5 5\theta} \]

### Step 2: Use the identity for cotangent
We can rewrite \( \tan^5 \theta \) in terms of cotangent:
\[ \tan^5 \theta = \cot^5 5\theta \]
This means:
\[ \tan^5 \theta = \cot^5(90^\circ - \theta) \]
Since \( \cot(90^\circ - x) = \tan x \), we can say:
\[ \tan^5 \theta = \tan^5(90^\circ - 5\theta) \]

### Step 3: Set up the equation
From the above, we can conclude:
\[ \theta = 90^\circ - 5\theta + n \cdot 180^\circ \quad (n \in \mathbb{Z}) \]
Solving for \( \theta \):
\[ 6\theta = 90^\circ + n \cdot 180^\circ \]
\[ \theta = 15^\circ + n \cdot 30^\circ \]
For the simplest case, we take \( n = 0 \):
\[ \theta = 15^\circ \]

### Step 4: Find \( \tan 3\theta \)
Now we need to find \( \tan 3\theta \):
\[ 3\theta = 3 \times 15^\circ = 45^\circ \]
Thus:
\[ \tan 3\theta = \tan 45^\circ = 1 \]

### Step 5: Calculate \( \tan^4 3\theta \)
Now we can find \( \tan^4 3\theta \):
\[ \tan^4 3\theta = \tan^4 45^\circ = 1^4 = 1 \]

### Final Answer
The value of \( \tan^4 3\theta \) is:
\[ \boxed{1} \]