If `theta` be acute angle and `tan(4theta - 50^(@) = cot(50^(@) - theta)`, then the value of `theta` in degrees

Options

A. 20
B. 50
C. 40
D. 30

Answer

To solve the equation \( \tan(4\theta - 50^\circ) = \cot(50^\circ - \theta) \), we can follow these steps:

### Step 1: Use the cotangent identity
Recall that \( \cot(x) = \tan(90^\circ - x) \). Therefore, we can rewrite the equation as:
\[ \tan(4\theta - 50^\circ) = \tan(90^\circ - (50^\circ - \theta)) \]
This simplifies to:
\[ \tan(4\theta - 50^\circ) = \tan(40^\circ + \theta) \]

### Step 2: Set the angles equal
Since the tangent function is periodic, we can set the angles equal to each other:
\[ 4\theta - 50^\circ = 40^\circ + \theta + n \cdot 180^\circ \quad (n \in \mathbb{Z}) \]
For acute angles, we will consider \( n = 0 \):
\[ 4\theta - 50^\circ = 40^\circ + \theta \]

### Step 3: Solve for \( \theta \)
Rearranging the equation gives:
\[ 4\theta - \theta = 40^\circ + 50^\circ \]
\[ 3\theta = 90^\circ \]
\[ \theta = \frac{90^\circ}{3} = 30^\circ \]

### Step 4: Conclusion
Thus, the value of \( \theta \) is:
\[ \theta = 30^\circ \]