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Jace Wright 2022-05-19 Answered
To prove: cot 1 7 + cot 1 8 + cot 1 18 = cot 1 3
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Answers (1)

Blaine Andrews
Answered 2022-05-20 Author has 20 answers
The statement is equivalent to
arccot 7 + arccot 8 = arccot 3 arccot 18
The right-hand side is positive because the arccotangent is decreasing, so it is in the interval ( 0 , π / 2 ) and we have to ensure that the same holds for the left-hand side, but this is easy, because 7>1 and 8>1, hence
arccot 7 + arccot 8 < 2 arccot 1 = π 2
The two sides are equal if and only if their cotangents are.
Since
cot ( α + β ) = cot α cot β 1 cot α + cot β cot ( α β ) = cot α cot β + 1 cot β cot α
this amounts to proving that
7 8 1 7 + 8 = 3 18 + 1 18 3
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