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mvcdc | Student, Graduate | (Level 2) Associate Educator

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We want to look for the minimum value of `cot^2A + cot^2B + cot^2C` given that `A + B + C = pi` .

`(cotA - cotB)^2 + (cotB - cotC)^2 + (cotC - cotA)^2 >= 0` since a square of any number is greater than or equal to zero (and the terms are just sum of the squares). Expanding:

`(cot^2A - 2cotAcotB + cot^2B + cot^2B - 2cotBcotC + cot^2C + cot^2C - 2cotCcotA + cot^2A >= 0`

This implies that:

`2cot^2A + 2cot^2B + 2cot^2C >= 2cotAcotB + 2cotBcotC + 2cotCcotA`

`cot^2A + cot^2B + cot^2C >= cotAcotB + cotBcotC + cotCcotA`

Hence, we simply need to look for the value of the sum on the right side to get the minimum.

We know that `A + B + C = pi` . Hence, `A + B = pi - C.`

Then, `tan(A + B) = tan(pi - C) = -tanC` by the property of the tangent. Using sum identities:

`tan(A + B) = (tanA+tanB)/(1-tanAtanB) = -tanC` which implies that:

`tanA + tanB + tanC = tanAtanBtanC` .

Dividing both sides by `tanAtanBtanC`  :

`(tanA)/(tanAtanBtanC) + (tanB)/(tanAtanBtanC) + (tanC)/(tanAtanBtanC) = 1`

`1/(tanBtanC) + 1/(tanAtanC) + 1/(tanAtanB) = 1`

`cotBcotC + cotAcotC + cotAcotB = 1`

Hence,

`cot^2A + cot^2B + cot^2C >= 1`

The minimum value is 1.