Suppose ABC is a right angled triangle with right angled at C . If length of sides opposite to angle A,B,C are respectively u,v and w then `tanA+tanB` equals

Options

A. `(u^(2))/(uv)`
B. `1`
C. `u+v`
D. `(w^(2))/(uv)`

Answer

To solve the problem, we need to find the value of \( \tan A + \tan B \) for a right-angled triangle ABC, where the right angle is at C, and the sides opposite to angles A, B, and C are u, v, and w respectively.

### Step-by-step Solution:

1. **Identify the Triangle and Angles**:
- We have a right-angled triangle ABC with right angle at C.
- The sides opposite to angles A, B, and C are u, v, and w respectively.
- Thus, \( AC = v \) (base), \( BC = u \) (perpendicular), and \( AB = w \) (hypotenuse).

2. **Write the Expressions for \( \tan A \) and \( \tan B \)**:
- The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side.
- For angle A:
\[ \tan A = \frac{\text{Opposite to A}}{\text{Adjacent to A}} = \frac{u}{v} \]
- For angle B:
\[ \tan B = \frac{\text{Opposite to B}}{\text{Adjacent to B}} = \frac{v}{u} \]

3. **Add the Tangents**:
- Now, we need to find \( \tan A + \tan B \):
\[ \tan A + \tan B = \frac{u}{v} + \frac{v}{u} \]

4. **Combine the Fractions**:
- To add these fractions, we need a common denominator:
\[ \tan A + \tan B = \frac{u^2 + v^2}{uv} \]

5. **Use the Pythagorean Theorem**:
- In a right triangle, by the Pythagorean theorem, we have:
\[ w^2 = u^2 + v^2 \]
- Thus, we can substitute \( u^2 + v^2 \) with \( w^2 \):
\[ \tan A + \tan B = \frac{w^2}{uv} \]

6. **Final Result**:
- Therefore, the final expression for \( \tan A + \tan B \) is:
\[ \tan A + \tan B = \frac{w^2}{uv} \]