If `angleA=90^(@)` in the `DeltaABC` , then `tan^(-1)((c)/(a+b))+tan^(-1)((b)/(a+c))` is equal to

Options

A. 0
B. 1
C. `(pi)/(4)`
D. `(pi)/(6)`

Explanation

To solve the problem, we need to evaluate the expression \( \tan^{-1}\left(\frac{c}{a+b}\right) + \tan^{-1}\left(\frac{b}{a+c}\right) \) given that \( \angle A = 90^\circ \) in triangle ABC.

### Step-by-Step Solution:

1. **Understanding the Triangle**:
Since \( \angle A = 90^\circ \), triangle ABC is a right triangle with \( A \) as the right angle. By convention, we denote the sides opposite to the angles as follows:
- Side \( a \) opposite angle \( A \)
- Side \( b \) opposite angle \( B \)
- Side \( c \) opposite angle \( C \)

2. **Using the Formula for Sum of Inverses**:
We use the formula for the sum of two inverse tangents:
\[ \tan^{-1}(x) + \tan^{-1}(y) = \tan^{-1}\left(\frac{x+y}{1-xy}\right) \]
where \( x = \frac{c}{a+b} \) and \( y = \frac{b}{a+c} \).

3. **Substituting Values**:
Substitute the values of \( x \) and \( y \):
\[ \tan^{-1}\left(\frac{c}{a+b}\right) + \tan^{-1}\left(\frac{b}{a+c}\right) = \tan^{-1}\left(\frac{\frac{c}{a+b} + \frac{b}{a+c}}{1 - \frac{c}{a+b} \cdot \frac{b}{a+c}}\right) \]

4. **Finding the Numerator**:
The numerator becomes:
\[ \frac{c(a+c) + b(a+b)}{(a+b)(a+c)} = \frac{ca + c^2 + ab + b^2}{(a+b)(a+c)} \]

5. **Finding the Denominator**:
The denominator becomes:
\[ 1 - \frac{cb}{(a+b)(a+c)} = \frac{(a+b)(a+c) - cb}{(a+b)(a+c)} = \frac{a^2 + ab + ac + bc - cb}{(a+b)(a+c)} = \frac{a^2 + ab + ac}{(a+b)(a+c)} \]

6. **Combining the Results**:
Now we can combine the results:
\[ \tan^{-1}\left(\frac{ca + c^2 + ab + b^2}{a^2 + ab + ac}\right) \]

7. **Using Pythagorean Theorem**:
Since \( A \) is a right angle, we can use the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
Thus, we can simplify the expression further.

8. **Final Simplification**:
After simplification, we find that:
\[ \tan^{-1}(1) = \frac{\pi}{4} \]

### Final Result:
Thus, the value of \( \tan^{-1}\left(\frac{c}{a+b}\right) + \tan^{-1}\left(\frac{b}{a+c}\right) \) is:
\[ \frac{\pi}{4} \]