If `tan ((theta)/(2))tan((2theta)/(5))=1` , what is the value (in degrees ) of `theta` ?

Options

A. `45^(@)`
B. `90^(@)`
C. `100^(@)`
D. `120^(@)`

Answer

To solve the equation \( \tan\left(\frac{\theta}{2}\right) \tan\left(\frac{2\theta}{5}\right) = 1 \), we can follow these steps:

### Step 1: Rewrite the equation using cotangent
We know that \( \tan A = \frac{1}{\cot A} \). Therefore, we can rewrite the equation as:
\[ \tan\left(\frac{\theta}{2}\right) = \cot\left(\frac{2\theta}{5}\right) \]
This implies:
\[ \tan\left(\frac{\theta}{2}\right) = \tan\left(90^\circ - \frac{2\theta}{5}\right) \]

### Step 2: Set the angles equal
Since the tangents are equal, we can set the angles equal to each other:
\[ \frac{\theta}{2} = 90^\circ - \frac{2\theta}{5} \]

### Step 3: Solve for \(\theta\)
To solve for \(\theta\), we first eliminate the fractions by finding a common denominator. The common denominator for 2 and 5 is 10. Multiply the entire equation by 10:
\[ 10 \cdot \frac{\theta}{2} = 10 \cdot \left(90^\circ - \frac{2\theta}{5}\right) \]
This simplifies to:
\[ 5\theta = 900^\circ - 4\theta \]

### Step 4: Combine like terms
Now, add \(4\theta\) to both sides:
\[ 5\theta + 4\theta = 900^\circ \]
This gives:
\[ 9\theta = 900^\circ \]

### Step 5: Divide to find \(\theta\)
Now, divide both sides by 9:
\[ \theta = \frac{900^\circ}{9} = 100^\circ \]

### Final Answer
Thus, the value of \(\theta\) is:
\[ \theta = 100^\circ \]