If a^2, b^2, c^2 are in AP. prove that cotA, cotB, cotC are in AP.

Given:a  

2

,b  

2

,c  

2

 are in A.P

⇒b  

2

−a  

2

=c  

2

−b  

2

⇒k  

2

sin  

2

B−k  

2

sin  

2

A=k  

2

sin  

2

C−k  

2

sin  

2

B using sine rule a=ksinA,b=ksinB and c=ksinC

⇒sin(B+A)sin(B−A)=sin(C+B)sin(C−B)

⇒sin(π−C)sin(B−A)=sin(π−A)sin(C−B) since A+B+C=π

⇒sinCsin(B−A)=sinAsin(C−B)

⇒  

sinA

sin(B−A)

​

=  

sinC

sin(C−B)

​

⇒  

sinAsinB

sin(B−A)

​

=  

sinBsinC

sin(C−B)

​

⇒  

sinAsinB

sinBcosA−cosBsinA

​

=  

sinBsinC

sinCcosB−cosBsinC

​

⇒cotA−cotB=cotB−cotC

∴cotA,cotB,cotC are in A.P

Step-by-step explanation:

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step-by-step explanation:

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