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Question

ABC is a triangle. Then tan2A2+tan2B2+tan2C2

A
>1
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B
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C
1
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D
1
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Solution

The correct option is C 1
A+B+C=180oC=180o(A+B)C2=π2A+B2tan(C/2)=tan(π/2(A+B)/2)=1tan((A+B)/2)=1tan(A/2)tan(B/2)tan(A/2)+tan(B/2)

Let a=tan(A/2),b=tan(B/2),c=tan(C/2) all positive, the constraint becomes
c=(1ab)/(a+b)

which is equivalent to ab+bc+ca=1

and we know that,

a2+b2+c2ab+bc+caa2+b2+c21tan2(A/2)+tan2(B/2)+tan2(C/2)1

Therefore, Answer is 1

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