In a triangle ABC, right angled at C, find the value of tan A + tan B in terms of the sides a,b,c.

Answer

To solve the problem of finding the value of \( \tan A + \tan B \) in terms of the sides \( a, b, c \) of triangle \( ABC \) (where \( C \) is the right angle), we can follow these steps:

### Step 1: Understand the triangle and its sides
In triangle \( ABC \):
- \( C \) is the right angle.
- The side opposite angle \( A \) is \( a \).
- The side opposite angle \( B \) is \( b \).
- The hypotenuse (the side opposite the right angle) is \( c \).

### Step 2: Write the expressions for \( \tan A \) and \( \tan B \)
Using the definitions of tangent in a right triangle:
- \( \tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AC} = \frac{a}{b} \)
- \( \tan B = \frac{\text{opposite}}{\text{adjacent}} = \frac{AC}{BC} = \frac{b}{a} \)

### Step 3: Add the two tangent values
Now, we can add \( \tan A \) and \( \tan B \):
\[ \tan A + \tan B = \frac{a}{b} + \frac{b}{a} \]

### Step 4: Find a common denominator
To add these fractions, we need a common denominator:
\[ \tan A + \tan B = \frac{a^2}{ab} + \frac{b^2}{ab} = \frac{a^2 + b^2}{ab} \]

### Step 5: Use the Pythagorean theorem
According to the Pythagorean theorem, in a right triangle:
\[ a^2 + b^2 = c^2 \]
Thus, we can substitute \( c^2 \) for \( a^2 + b^2 \):
\[ \tan A + \tan B = \frac{c^2}{ab} \]

### Final Result
Therefore, the value of \( \tan A + \tan B \) in terms of the sides \( a, b, c \) is:
\[ \tan A + \tan B = \frac{c^2}{ab} \]

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