`DeltaDEF` is right angled at E .If `mangleD=45^(@)` , then find the value of `(tanF+(1)/(3))` .

Options

A. `(4)/(3)`
B. `(3sqrt(3))/(2)`
C. `((sqrt(2)+1))/(sqrt(2))`
D. `((3sqrt(2)+1))/(3)`

Answer

To solve the problem, we need to find the value of \( \tan F + \frac{1}{3} \) in triangle \( \Delta DEF \), which is right-angled at \( E \) and has \( \angle D = 45^\circ \).

### Step-by-Step Solution:

1. **Identify the Angles in Triangle DEF**:
- Since \( \Delta DEF \) is a right-angled triangle at \( E \), we know that \( \angle E = 90^\circ \).
- We are given that \( \angle D = 45^\circ \).
- We need to find \( \angle F \).

2. **Calculate Angle F**:
- The sum of angles in a triangle is \( 180^\circ \).
- Therefore, we can find \( \angle F \) using the formula:
\[ \angle F = 180^\circ - (\angle D + \angle E) \]
- Substituting the known values:
\[ \angle F = 180^\circ - (45^\circ + 90^\circ) = 180^\circ - 135^\circ = 45^\circ \]

3. **Calculate \( \tan F \)**:
- Since \( \angle F = 45^\circ \), we can find \( \tan F \):
\[ \tan F = \tan 45^\circ = 1 \]

4. **Find \( \tan F + \frac{1}{3} \)**:
- Now we can substitute \( \tan F \) into the expression:
\[ \tan F + \frac{1}{3} = 1 + \frac{1}{3} \]
- To add these, we need a common denominator:
\[ 1 = \frac{3}{3} \]
- Therefore:
\[ \tan F + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \]

5. **Final Answer**:
- The value of \( \tan F + \frac{1}{3} \) is:
\[ \frac{4}{3} \]