Question

if a+b+c=π,find cota.cotb+cotb.cotc+cotc.cota

Answer 1

Sum of all angles of triangle is 180.⇒A+B+C=180⇒A+B=180−CTaking tan on both sides⇒tan(A+B)=tan(180−C)⇒tan(A)+tan(B)1−tan(A).tan(B)=−tan(C)⇒tan(A)+tan(B)=−tan(C)+tan(A).tan(B)tan(C)⇒tan(A)+tan(B)+tan(C)=tan(A).tan(B)tan(C)Dividing each term  by tan(A).tan(B)tan(C) we get:tan(A)tan(A).tan(B)tan(C)+tan(B)tan(A).tan(B)tan(C)+tan(C)tan(A).tan(B)tan(C)=tan(A).tan(B)tan(C)tan(A).tan(B)tan(C)⇒1tan(B)tan(C)+1tan(A).tan(C)+1tan(A).tan(B)=1⇒cot(B).cot(C)+cot(A).cot(C)+cot(A).cot(B)=1⇒cot(A).cot(B)+cot(B).cot(C)+cot(C).cot(A)=1
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