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:

i believe the correct answer is option 4.principalities were small territories that were run by russian rulers.

which statement correctly defines principalities in northern asia in the 1200s? principalities were small territories that were run by russian rulers.

hope i could ! : )