Question:
If tan A + tan B + tan C = 6, find the value of cot A * cot B * cot C
Answer:
In a triangle, sum of all the angles is equal to 180 degrees
=> A + B + C = 180
Now, tan(A + B + c) = tan 180
=> (tan A + tan B + tan C - tan A * tan B * tan C)/{1 - (tan A * tan B + tan B * tan C
+ tan A * tan C)} = 0
=> tan A + tan B + tan C - tan A * tan B * tan C = 0
=> 6 - tan A * tan B * tan C = 0
=> tan A * tan B * tan C = 6
=> (1/cot A) * (1/cot B) * (1/cot C) = 6
=> 1/(cot A * cot B * cot C) = 6
=> cot A * cot B * cot C = 1/6
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