Prove that: `cot x + cot (60+x) - cot (60-x) = 3 cot 3x`

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justaguide | College Teacher | (Level 2) Distinguished Educator

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It has to be shown that ` cot x + cot (60+x) - cot (60-x) = 3 cot 3x`

`cot x + cot (60+x) - cot (60-x)`

Use the formula: `tan(a + b) = (1 - tan a*tan b)/(tan a + tan b)`

=>` cot x + (1 - tan 60*tan x)/(tan 60 + tan x) - (1 + tan 60*tan x)/(tan 60 - tan x)`

=> `1/tan x + (1 - sqrt 3*tan x)/(sqrt 3 + tan x) - (1 + sqrt 3*tan x)/(sqrt 3 - tan x)`

=> `(3 - tan^2x + sqrt3*tan x - tan^2x - sqrt3*tanx - tan^2x)/(tanx*(3 - tan^2x))`

=> `(3 - 3*tan^2x)/(tanx*(3 - tan^2x))`

=> `(3*(1 - tan^2x))/(tanx*(3 - tan^2x))`

Use the identity `tan 3x = ((3tanx-tan^3x) / (1-3tan^2x))`

=> `3/tan(3x)`

=> `3*cot 3x`

This proves that `cot x + cot (60+x) - cot (60-x) = 3 cot 3x`

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