Trigonometry> If A+B+C =180, then find the minimum valu...

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

Aakash , 11 Years ago

Grade 12th pass

3 Answers

Sunil Raikwar

Last Activity: 11 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

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Jaikarthik Suresh

Last Activity: 8 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

Last Activity: 7 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.