Let triangle ABC be right angled at C.If `tanA+tanB=2`, then-

Options

A. `/_A = 30^(@)`
B. triangle ABC must be isoscles
C. sides of the triangle ABC are 3,4,5
D. triangle must be scalene

Answer

To solve the problem, we need to analyze the given information about triangle ABC, which is right-angled at C. We are given that \( \tan A + \tan B = 2 \). Let's go through the solution step by step.

### Step 1: Understand the Angles in Triangle ABC
In a triangle, the sum of the angles is \( 180^\circ \). Since triangle ABC is right-angled at C, we have:
\[ A + B + C = 180^\circ \]
Given that \( C = 90^\circ \), we can simplify this to:
\[ A + B = 90^\circ \]

**Hint:** Remember that in a right-angled triangle, the two non-right angles are complementary.

### Step 2: Express \( B \) in Terms of \( A \)
From the equation \( A + B = 90^\circ \), we can express angle \( B \) as:
\[ B = 90^\circ - A \]

**Hint:** Use the complementary angle relationship to express one angle in terms of the other.

### Step 3: Substitute \( B \) into the Tangent Equation
We know that:
\[ \tan A + \tan B = 2 \]
Substituting \( B = 90^\circ - A \) into the equation, we get:
\[ \tan A + \tan(90^\circ - A) = 2 \]
Using the identity \( \tan(90^\circ - A) = \cot A \), we can rewrite the equation as:
\[ \tan A + \cot A = 2 \]

**Hint:** Recall the trigonometric identity that relates tangent and cotangent.

### Step 4: Rewrite \( \cot A \) in Terms of \( \tan A \)
We know that \( \cot A = \frac{1}{\tan A} \). Thus, we can rewrite the equation:
\[ \tan A + \frac{1}{\tan A} = 2 \]

**Hint:** Use the relationship between cotangent and tangent to simplify the equation.

### Step 5: Multiply Through by \( \tan A \)
Let \( x = \tan A \). The equation becomes:
\[ x + \frac{1}{x} = 2 \]
Multiplying through by \( x \) gives:
\[ x^2 + 1 = 2x \]
Rearranging this gives us a quadratic equation:
\[ x^2 - 2x + 1 = 0 \]

**Hint:** Form a quadratic equation by eliminating the fraction.

### Step 6: Factor the Quadratic Equation
The quadratic equation can be factored as:
\[ (x - 1)^2 = 0 \]
This implies:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]

**Hint:** Recognize that a perfect square indicates a double root.

### Step 7: Find \( \tan A \) and \( \tan B \)
Since \( x = \tan A \), we have:
\[ \tan A = 1 \]
Thus, angle \( A \) is:
\[ A = 45^\circ \]
Using \( B = 90^\circ - A \):
\[ B = 90^\circ - 45^\circ = 45^\circ \]

**Hint:** Use the inverse tangent function to find the angle from its tangent value.

### Step 8: Conclude the Properties of Triangle ABC
Since both angles \( A \) and \( B \) are equal, triangle ABC is an isosceles right triangle.

**Final Conclusion:** Triangle ABC must be isosceles.

### Final Answer
Triangle ABC is isosceles.