If `tan 4 theta = cot (2 theta +30^(@))`, then `theta` is equal to :

Options

A. `15^(@)`
B. `10^(@)`
C. `20^(@)`
D. `25^(@)`

Answer

To solve the equation \( \tan 4\theta = \cot(2\theta + 30^\circ) \), we can follow these steps:

### Step 1: Rewrite the cotangent function
We know that \( \cot x = \tan(90^\circ - x) \). Therefore, we can rewrite the equation as:
\[ \tan 4\theta = \tan(90^\circ - (2\theta + 30^\circ)) \]

### Step 2: Simplify the right side
Now, simplify the right side:
\[ 90^\circ - (2\theta + 30^\circ) = 90^\circ - 2\theta - 30^\circ = 60^\circ - 2\theta \]
Thus, we have:
\[ \tan 4\theta = \tan(60^\circ - 2\theta) \]

### Step 3: Set the angles equal
Since the tangent function is periodic, we can set the angles equal to each other:
\[ 4\theta = 60^\circ - 2\theta + n \cdot 180^\circ \quad \text{(where \( n \) is any integer)} \]

### Step 4: Solve for \( \theta \)
For the simplest case, let \( n = 0 \):
\[ 4\theta + 2\theta = 60^\circ \]
\[ 6\theta = 60^\circ \]
\[ \theta = \frac{60^\circ}{6} = 10^\circ \]

### Step 5: Conclusion
Thus, the value of \( \theta \) is:
\[ \theta = 10^\circ \]