If A, B, C are the angles of a triangle such that `sin^(2)A+sin^(2)B=sin^(2)C`, then

Options

A. sin A + sin B >1
B. tan A tan B = 1
C. sin A + sin B = 1
D. tan A. tan B < 1

Explanation

To solve the problem, we start with the given equation:

\[ \sin^2 A + \sin^2 B = \sin^2 C \]

### Step 1: Use the Sine Rule
According to the sine rule in a triangle, we have:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k \]

From this, we can express \(\sin A\), \(\sin B\), and \(\sin C\) in terms of the sides of the triangle:

\[ \sin A = \frac{a}{k}, \quad \sin B = \frac{b}{k}, \quad \sin C = \frac{c}{k} \]

### Step 2: Substitute into the Given Equation
Now substitute these values into the equation \(\sin^2 A + \sin^2 B = \sin^2 C\):

\[ \left(\frac{a}{k}\right)^2 + \left(\frac{b}{k}\right)^2 = \left(\frac{c}{k}\right)^2 \]

This simplifies to:

\[ \frac{a^2}{k^2} + \frac{b^2}{k^2} = \frac{c^2}{k^2} \]

### Step 3: Eliminate \(k^2\)
Since \(k^2\) is common in all terms, we can multiply through by \(k^2\) (assuming \(k \neq 0\)):

\[ a^2 + b^2 = c^2 \]

### Step 4: Identify the Triangle Type
The equation \(a^2 + b^2 = c^2\) is the Pythagorean theorem, which indicates that triangle ABC is a right triangle with angle \(C\) being \(90^\circ\).

### Step 5: Use Angle Sum Property
Since angle \(C\) is \(90^\circ\), we can use the angle sum property of triangles:

\[ A + B + C = 180^\circ \]

Substituting \(C = 90^\circ\):

\[ A + B + 90^\circ = 180^\circ \]

Thus, we have:

\[ A + B = 90^\circ \]

### Step 6: Find \(\tan A + B\)
Taking the tangent of both sides:

\[ \tan(A + B) = \tan(90^\circ) \]

Since \(\tan(90^\circ)\) is undefined (or infinite), we can use the tangent addition formula:

\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]

Setting this equal to infinity implies:

\[ 1 - \tan A \tan B = 0 \quad \Rightarrow \quad \tan A \tan B = 1 \]

### Step 7: Conclusion
Thus, we conclude that:

1. \(\tan A \tan B = 1\)
2. \(\sin A + \sin B = 1\) (since \(\sin C = 1\) when \(C = 90^\circ\))

### Final Answer
The correct options from the given choices are:

- \(\tan A \tan B = 1\) (Option 2)
- \(\sin A + \sin B = 1\) (Option 3)