In an isosceles right angled triangle , a straight line drwan from the mid - point of one of equal sides to the opposite angle . It divides the angle into two parts , `theta and (pi//4-theta)` . Then `tan theta and tan [(pi//4) -theta]` are equal to

Options

A. `1/2,1/3`
B. `1/3,1/4`
C. `1/5,1/6`
D. none of these

Answer

To solve the problem, we need to find the values of \( \tan \theta \) and \( \tan \left( \frac{\pi}{4} - \theta \right) \) in the context of an isosceles right triangle where a line is drawn from the midpoint of one of the equal sides to the opposite angle.

### Step-by-Step Solution:

1. **Define the Triangle**:
Let's denote the isosceles right triangle as \( \triangle ABC \) where \( AB = AC = 2x \) and \( \angle A = 90^\circ \). The midpoint of side \( AB \) is point \( G \).

2. **Draw the Line**:
Draw a line from point \( G \) to point \( C \). This line divides \( \angle A \) into two angles: \( \theta \) and \( \frac{\pi}{4} - \theta \).

3. **Identify Triangle \( \triangle ABG \)**:
In triangle \( ABG \):
- The length \( AG = x \) (since \( G \) is the midpoint of \( AB \)).
- The length \( BG = x \) (since \( G \) is the midpoint of \( AB \)).
- The length \( AB = 2x \).

4. **Calculate \( \tan \theta \)**:
Using the definition of tangent in triangle \( ABG \):
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{AG}{BG} = \frac{x}{x} = 1 \]
However, we need to find the relationship involving \( 2x \) and \( x \):
\[ \tan \theta = \frac{2x}{x} = 2 \quad \text{(using the full length of the side)} \]

5. **Calculate \( \tan \left( \frac{\pi}{4} - \theta \right) \)**:
We use the identity:
\[ \tan \left( \frac{\pi}{4} - \theta \right) = \frac{1 - \tan \theta}{1 + \tan \theta} \]
Substituting \( \tan \theta = 2 \):
\[ \tan \left( \frac{\pi}{4} - \theta \right) = \frac{1 - 2}{1 + 2} = \frac{-1}{3} = -\frac{1}{3} \]

6. **Final Values**:
Thus, we have:
\[ \tan \theta = 2 \quad \text{and} \quad \tan \left( \frac{\pi}{4} - \theta \right) = -\frac{1}{3} \]

### Conclusion:
The values of \( \tan \theta \) and \( \tan \left( \frac{\pi}{4} - \theta \right) \) are \( 2 \) and \( -\frac{1}{3} \) respectively.