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Question
In any ΔABC if a2 , b2 , c2 are in arithmetic progression, then prove that Cot A, Cot B, Cot C are in arithmetic progression.
Solution
Given that a2 ,b2 ,c2 are in arithmetic progression.
We need to prove that cotA, cotB and cotC are in
arithmetic progression.
a2 ,b2 ,c2 are in A.P.
`-2a^2, -2b^2, -2c^2 " are in A.P"`
`(a^2+b^2+c^2)-2a^2,(a^2+b^2+c^2)-2b^2, (a^2+b^2+c^2)-2c^2 " are in A.P"`
`(b^2+c^2-a^2), (c^2+a^2-b^2),(a^2+b^2-c^2) " are in A.P "`
`(b^2+c^2-a^2)/(2abc), (c^2+a^2-b^2)/(2abc),(a^2+b^2-c^2) /(2abc)" are in A.P "`
`1/a(b^2+c^2-a^2)/(2bc), 1/b(c^2+a^2-b^2)/(2ac),1/c(a^2+b^2-c^2) /(2ab)" are in A.P "`
`1/acosA,1/bcosB,1/c cos C " are in A.P"`
`k/acosA,k/bcosB,k/c cos C " are in A.P"`
`cosA/sinA,cosB/sinB,cosC/sinC " are in A.P"`
`cotA,cotB,cotC " are in A.P"`
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