If tan `4theta = cot(40^@ – 2theta)`, then `theta` is equal to:

Options

A. `20^@`
B. `35^@`
C. `25^@`
D. `30^@`

Answer

To solve the equation \( \tan 4\theta = \cot(40^\circ - 2\theta) \), we can follow these steps:

### Step 1: Rewrite the cotangent
We know that \( \cot x = \tan(90^\circ - x) \). Therefore, we can rewrite the equation as:
\[ \tan 4\theta = \tan(90^\circ - (40^\circ - 2\theta)) \]

### Step 2: Simplify the right side
Now simplify the right side:
\[ \tan 4\theta = \tan(90^\circ - 40^\circ + 2\theta) = \tan(50^\circ + 2\theta) \]

### Step 3: Set the angles equal
Since the tangent function is periodic, we can set the angles equal to each other:
\[ 4\theta = 50^\circ + 2\theta \]

### Step 4: Solve for \( \theta \)
Now, we will isolate \( \theta \):
\[ 4\theta - 2\theta = 50^\circ \]
\[ 2\theta = 50^\circ \]
\[ \theta = 25^\circ \]

### Conclusion
Thus, the value of \( \theta \) is:
\[ \theta = 25^\circ \]