# If A+B+C =180, then find the minimum value of cot^2(A)+cot^2(B)+cot^2(C).

Sunil Raikwar
askIITians Faculty 45 Points
10 years ago
We knoe that for positive integers
AM greater than equal to GM
cot2A+cot2B/2greater than equal to square root of (cot2Acot2B)
cot2A+cot2B greater than equal to 2cotAcotB.............1
cot2B+cot2Cgreater than equal to 2cotBcotC............2
cot2A+cot2Cgreater than equal to 2cotAcotC.............3
adding eq. 1, 2 & 3
cot2A+cot2B+cot2c greater than equal to 2(cotAcotB+cotBcotC+cotCcotA)
if A+B+C=180
then cotAcotB+cotBcotC+cotCcotA=1
cot2A+cot2B+cot2cgreater than equal to 2
Thanks & Regards
Sunil Raikwar
Jaikarthik Suresh
24 Points
7 years ago
We knoe that for positive integersAM greater than equal to GMcot2A+cot2B/2greater than equal to square root of (cot2Acot2B)cot2A+cot2B greater than equal to 2cotAcotB.............1cot2B+cot2Cgreater than equal to 2cotBcotC............2cot2A+cot2Cgreater than equal to 2cotAcotC.............3adding eq. 1, 2 & 32(cot2A+cot2B+cot2c) greater than equal to 2(cotAcotB+cotBcotC+cotCcotA)if A+B+C=180then cotAcotB+cotBcotC+cotCcotA=1cot2A+cot2B+cot2cgreater than equal to 1
Shrey Patel
11 Points
6 years ago
We know that for positive integers AM greater than equal to GMcot2A+cot2B/2 greater than equal to square root of (cot2Acot2B)cot2A+cot2B greater than equal to 2cotAcotB.............1cot2B+cot2C greater than equal to 2cotBcotC............2cot2A+cot2C greater than equal to 2cotAcotC.............3adding eq. 1, 2 & 3cot2A+cot2B+cot2c greater than equal to cotAcotB+cotBcotC+cotCcotAif A+B+C=180then cotAcotB+cotBcotC+cotCcotA=1cot2A+cot2B+cot2c greater than equal to 1Hence, answer is 1.