If tan `((theta)/(2)) tan ((2 theta)/(5))` =1 then 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 will follow these steps:

### Step 1: Use the identity for tangent addition
We know that if \( \tan A \tan B = 1 \), then \( A + B = 90^\circ \). In our case, let:
- \( A = \frac{\theta}{2} \)
- \( B = \frac{2\theta}{5} \)

Thus, we can write:
\[ \frac{\theta}{2} + \frac{2\theta}{5} = 90^\circ \]

### Step 2: Find a common denominator
To combine the fractions, we need a common denominator. The least common multiple of 2 and 5 is 10. Therefore, we rewrite the equation:
\[ \frac{5\theta}{10} + \frac{4\theta}{10} = 90^\circ \]

### Step 3: Combine the fractions
Now we can combine the fractions:
\[ \frac{5\theta + 4\theta}{10} = 90^\circ \]
This simplifies to:
\[ \frac{9\theta}{10} = 90^\circ \]

### Step 4: Solve for \( \theta \)
To isolate \( \theta \), multiply both sides by 10:
\[ 9\theta = 900^\circ \]
Now divide by 9:
\[ \theta = 100^\circ \]

### Conclusion
The value of \( \theta \) is \( 100^\circ \).

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