Multiple choice

If $A,B,C$ are the angles of $\triangle ABC$ then $\cot { A } .\cot { B } +\cot { B } .\cot { C } +\cot { C } .\cot { A } $ is equal to

  1. $0$
  2. $1$
  3. $2$
  4. $-1$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In any triangle ABC, the sum of the angles is A + B + C = 180 degrees. Using the trigonometric identity for the sum of three angles, we have tan A + tan B + tan C = tan A * tan B * tan C. Dividing this entire equation by the product tan A * tan B * tan C yields the identity cot B * cot C + cot A * cot C + cot A * cot B = 1.

AI explanation

Because A, B, and C are the angles of a triangle, the angle sum property gives A + B = 180 - C. Taking the cotangent of both sides results in cot(A + B) = cot(180 - C), which expands using the cotangent addition formula to (cot A cot B - 1) / (cot B + cot A) = -cot C. Multiplying by (cot B + cot A) yields cot A cot B - 1 = -cot A cot C - cot B cot C. Moving all the product terms to the left side gives cot A cot B + cot B cot C + cot C cot A = 1.

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