If A,B,C are the angles of a triangle such that C is an obtuse angle then

Options

A. `tan A tan B lt 1`
B. `tan A tan B gt1`
C. `tan A tan B=1`
D. None

Answer

To solve the problem, we need to analyze the angles A, B, and C of a triangle where C is an obtuse angle.

### Step-by-Step Solution:

1. **Understanding the Angles of a Triangle**:
In any triangle, the sum of the angles is always 180 degrees. Therefore, we have:
\[ A + B + C = 180^\circ \]

2. **Identifying the Range of C**:
Since C is given as an obtuse angle, we know:
\[ 90^\circ < C < 180^\circ \]
This implies that:
\[ A + B < 90^\circ \]

3. **Using the Tangent Function**:
We can use the tangent function for the angles A, B, and C. The identity we will use is:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Since \(A + B = 180^\circ - C\), we can express this as:
\[ \tan(180^\circ - C) = -\tan C \]

4. **Setting Up the Inequality**:
From the tangent addition formula, we can derive:
\[ -\tan C = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Since C is obtuse, \(\tan C < 0\). Therefore, we can conclude that:
\[ \tan A + \tan B < 0 \quad \text{(since the denominator is positive)} \]

5. **Analyzing the Product**:
From the above, we can also analyze the product:
\[ \tan A \tan B < 1 \]
This leads us to conclude that:
\[ \tan A \tan B < 1 \]

6. **Final Conclusion**:
Thus, we can summarize that if C is an obtuse angle in triangle ABC, then:
\[ \tan A + \tan B < \tan A \tan B \]
This inequality holds true under the given conditions.