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If A+B+C=pi, prove that tan^2A/2+tan^2B/...

If `A+B+C=pi,` prove that `tan^2A/2+tan^2B/2+tan^2C/2geq1.`

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`(tan""(A)/(2)-tan""(B)/(2))^(2)+(tan""(B)/(2)-tan""(C)/(2))^(2)+(tan""(C)/(2)-tan""(A)/(2))^(2)ge0`
`therefore 2(tan^(2)""(A)/(2)+tan^(2)""(B)/(2)+tan^(2)""(C)/(2))`
`-2(tan""(A)/(2)tan""(B)/(2)+tan""(C)/(2)tan""(A)/(2)tan""(C)/(2))ge0`
we know that in `triangle ABC`,
`tan ""(A)/(2)tan""(B)/(2)+tan""(B)/(2)tan""(C)/(2)+tan""(A)/(2)tan ""(C)/(2)=1`
`therefore 2(tan^(2)""(A)/(2)+tan^(2)""(B)/(2)+tan^(2)""(C)/(2))-2(1)ge0`
`therefore tan^(2)""(A)/(2)+tan^(2)""(B)/(2)+tan^(2)""(C)/(2)ge1`
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