Find other 4-tuples of distinct integers between and that

satisfy the relation

Other solutions:

How many such equations can you produce if we allow repeated angles?

Hint: there are 49 such equations

Determine whether there are 3-term equations and 5-term equations.

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Here are the 4 term ones with distinct integers

Tan 6 + Tan 36 + Tan 66 = Tan 72.

Tan 10 + Tan 40 + Tan 60 = Tan 70.

Tan 24 + Tan 42 + Tan 60 = Tan 72.

Tan 50 + Tan 60 + Tan 70 = Tan 80.

Tan 60 + Tan 72 + Tan 78 = Tan 84.

and those with repeats

Tan 4 + Tan 4 + Tan 43 = Tan 47.

Tan 4 + Tan 82 + Tan 82 = Tan 86.

Tan 5 + Tan 80 + Tan 80 = Tan 85.

Tan 6 + Tan 6 + Tan 42 = Tan 48.

Tan 6 + Tan 36 + Tan 66 = Tan 72.

Tan 6 + Tan 78 + Tan 78 = Tan 84.

Tan 7 + Tan 76 + Tan 76 = Tan 83.

Tan 8 + Tan 8 + Tan 41 = Tan 49.

Tan 8 + Tan 74 + Tan 74 = Tan 82.

Tan 9 + Tan 72 + Tan 72 = Tan 81.

Tan 10 + Tan 10 + Tan 40 = Tan 50.

Tan 10 + Tan 40 + Tan 60 = Tan 70.

Tan 11 + Tan 68 + Tan 68 = Tan 79.

Tan 13 + Tan 64 + Tan 64 = Tan 77.

Tan 14 + Tan 14 + Tan 38 = Tan 52.

Tan 14 + Tan 62 + Tan 62 = Tan 76.

Tan 16 + Tan 58 + Tan 58 = Tan 74.

Tan 17 + Tan 56 + Tan 56 = Tan 73.

Tan 18 + Tan 18 + Tan 36 = Tan 54.

Tan 18 + Tan 54 + Tan 54 = Tan 72.

Tan 19 + Tan 52 + Tan 52 = Tan 71.

Tan 20 + Tan 20 + Tan 35 = Tan 55.

Tan 20 + Tan 50 + Tan 50 = Tan 70.

Tan 22 + Tan 46 + Tan 46 = Tan 68.

Tan 23 + Tan 44 + Tan 44 = Tan 67.

Tan 24 + Tan 42 + Tan 42 = Tan 66.

Tan 24 + Tan 42 + Tan 60 = Tan 72.

Tan 25 + Tan 40 + Tan 40 = Tan 65.

Tan 26 + Tan 38 + Tan 38 = Tan 64.

Tan 27 + Tan 36 + Tan 36 = Tan 63.

Tan 28 + Tan 34 + Tan 34 = Tan 62.

Tan 29 + Tan 32 + Tan 32 = Tan 61.

Tan 50 + Tan 60 + Tan 70 = Tan 80.

Tan 60 + Tan 72 + Tan 78 = Tan 84.

There are no solutions with 2 or 5 terms (0 < Integers <= 90)

Paul.

I also found

giving us seven 4-term (A, B, C, D) of distinct integers

and those with repeats, I found 49 such equations.

True. There are no such 3-term equations and no such 5-term equations.

For those with repeats indicated by Paul satisfy the relation

Tan x + Tan x + Tan y = Tan (x+y)