In a triangle `A B C ,/_C=pi/2dot` If `tan(A/2)a n dtan(B/2)` are the roots of the equation `a x^2+b x+c=0,(a!=0),` then the value of `(a+b)/c` (where `a , b , c ,` are sides of `` opposite to angles `A , B , C ,` respectively) is

Explanation

To solve the problem, we will follow the steps outlined in the video transcript and derive the required value step by step.

### Step 1: Understanding the Triangle and Roots
Given a triangle \( ABC \) where \( \angle C = \frac{\pi}{2} \) (90 degrees), we know that:
- The roots of the equation \( ax^2 + bx + c = 0 \) are \( \tan\left(\frac{A}{2}\right) \) and \( \tan\left(\frac{B}{2}\right) \).

### Step 2: Using the Properties of Roots
From the properties of quadratic equations, if \( \alpha \) and \( \beta \) are the roots, then:
- \( \alpha + \beta = -\frac{b}{a} \)
- \( \alpha \beta = \frac{c}{a} \)

Substituting the roots:
- \( \tan\left(\frac{A}{2}\right) + \tan\left(\frac{B}{2}\right) = -\frac{b}{a} \) (1)
- \( \tan\left(\frac{A}{2}\right) \tan\left(\frac{B}{2}\right) = \frac{c}{a} \) (2)

### Step 3: Using the Angle Sum Property
Since \( \angle A + \angle B + \angle C = \pi \) and \( \angle C = \frac{\pi}{2} \), we have:
\[ \angle A + \angle B = \frac{\pi}{2} \]
Thus,
\[ \frac{A + B}{2} = \frac{\pi}{4} \]

### Step 4: Finding the Tangent
Taking the tangent of both sides:
\[ \tan\left(\frac{A + B}{2}\right) = \tan\left(\frac{\pi}{4}\right) = 1 \]

Using the tangent addition formula:
\[ \tan\left(\frac{A}{2} + \frac{B}{2}\right) = \frac{\tan\left(\frac{A}{2}\right) + \tan\left(\frac{B}{2}\right)}{1 - \tan\left(\frac{A}{2}\right)\tan\left(\frac{B}{2}\right)} = 1 \]

### Step 5: Setting Up the Equation
From the tangent addition formula, we can equate:
\[ \frac{\tan\left(\frac{A}{2}\right) + \tan\left(\frac{B}{2}\right)}{1 - \tan\left(\frac{A}{2}\right)\tan\left(\frac{B}{2}\right)} = 1 \]
Cross-multiplying gives:
\[ \tan\left(\frac{A}{2}\right) + \tan\left(\frac{B}{2}\right) = 1 - \tan\left(\frac{A}{2}\right)\tan\left(\frac{B}{2}\right) \]

### Step 6: Substituting Values
Substituting equations (1) and (2) into the equation we derived:
\[ -\frac{b}{a} = 1 - \frac{c}{a} \]
Rearranging gives:
\[ -\frac{b}{a} + \frac{c}{a} = 1 \]
Multiplying through by \( a \):
\[ -c + b = a \]
Thus,
\[ a + b = c \]

### Step 7: Finding the Required Value
We need to find the value of:
\[ \frac{a + b}{c} \]
Substituting \( a + b = c \):
\[ \frac{c}{c} = 1 \]

### Final Answer
The value of \( \frac{a + b}{c} \) is \( \boxed{1} \).

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