The value of \( \cot{78} \cot{47} + \cot{55} \cot{47} + \cot{55} \cot{78} \) is:

A. 1

To solve the problem, we need to find the value of the expression:
cot 7 8 ∘ cot 4 7 ∘ + cot 5 5 ∘ cot 4 7 ∘ + cot 5 5 ∘ cot 7 8 ∘
Let's break it down using properties of cotangent and complementary angles:

We know that cot ( 9 0 ∘ − θ ) = tan ( θ ) . This means:

cot ( 7 8 ∘ ) = tan ( 1 2 ∘ ) because 7 8 ∘ = 9 0 ∘ − 1 2 ∘
cot ( 5 5 ∘ ) = tan ( 3 5 ∘ ) because 5 5 ∘ = 9 0 ∘ − 3 5 ∘
cot ( 4 7 ∘ ) = tan ( 4 3 ∘ ) because 4 7 ∘ = 9 0 ∘ − 4 3 ∘


The given expression becomes:
tan ( 1 2 ∘ ) tan ( 4 3 ∘ ) + tan ( 3 5 ∘ ) tan ( 4 3 ∘ ) + tan ( 3 5 ∘ ) tan ( 1 2 ∘ )

A useful identity involving tangents in a triangle is:

If the angles of a triangle are A , B , and C , then: tan ( A ) + tan ( B ) + tan ( C ) = tan ( A ) tan ( B ) tan ( C )

Here, we consider a triangle with angles 1 2 ∘ , 3 5 ∘ , and 4 3 ∘ , whose sum is 9 0 ∘ , forming a right triangle. The property is satisfied since tan ( 1 2 ∘ ) tan ( 3 5 ∘ ) tan ( 4 3 ∘ ) = 1 .



Using this identity:
tan ( 1 2 ∘ ) tan ( 4 3 ∘ ) + tan ( 3 5 ∘ ) tan ( 4 3 ∘ ) + tan ( 1 2 ∘ ) tan ( 3 5 ∘ ) = 1


So the value of the expression cot 78 cot 47 + cot 55 cot 47 + cot 55 cot 78 is 1 ​ . Therefore, the correct answer is option A. 1

Answered by BenjaminOwenLewis | 2025-07-06

The value of the expression cot 7 8 ∘ cot 4 7 ∘ + cot 5 5 ∘ cot 4 7 ∘ + cot 5 5 ∘ cot 7 8 ∘ is 1. This is derived using properties of cotangent and tangent in a triangle. The correct answer is option A. 1.
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Answered by BenjaminOwenLewis | 2025-07-10