If for a $\displaystyle \Delta ABC, \cot A.\cot B.\cot C> 0$ then the triangle is
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right angled
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acute angled
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obtuse angled
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all these options are possible
cot A * cot B * cot C > 0 implies either all three are positive (acute triangle) or one is positive and two are negative (obtuse triangle). However, for a triangle, the product of cotangents is positive only for acute triangles.
In any triangle, the sum of the angles is 180 degrees, so if one angle is obtuse, its cotangent is negative while the cotangents of the two acute angles are positive, making the overall product negative. If the triangle contains a 90 degree angle, the cotangent of that right angle is zero, making the product zero. Since the product of the cotangents is strictly greater than zero, neither an obtuse nor a right angle can exist within the triangle. Therefore, all three angles must be less than 90 degrees, making it an acute angled triangle.