Consider the following statements: 1. In a triangle ABC, if \cot A\cdot \cot B\cdot \cot C \gt 0, then the triangle is an acute angled triangle. 2. In a triangle ABC, if \tan A\cdot \tan B\cdot \tan C \gt 0, then the triangle is an obtuse angled triangle. Which of the statements given above is/are correct?
- A. 1 only ✓
- B. 2 only
- C. Both 1 and 2
- D. Neither 1 nor 2
Correct Answer: A. 1 only
Explanation
In any triangle, \tan A + \tan B + \tan C = \tan A \tan B \tan C. If all angles are acute, the product of their tangents is positive. If one angle is obtuse, its tangent is negative, making the product negative. Thus, \tan A \tan B \tan C \gt 0 indicates an acute-angled triangle, making Statement 2 false. The condition \cot A \cot B \cot C \gt 0 implies the product of their tangents is also positive, which correctly indicates an acute triangle, making Statement 1 true.