In a ` A B C ,` if `tanA/2,tanB/2,tanC/2a r einAdotPdot,` then show that `cosA ,cosB ,cosC` are in `AdotPdot`

Explanation

Given `tan""A//2,tanB//2,tanC//2` are in AP .
`therefore tanA//2-tanB//2=tanB//2-tanC//2`
`therefore (sinA//2)/(cosA//2)-(sinB//2)/(cosB//2)=(sinB//2)/(cosB//2)-(sinC//2)/(cosC//2)`
`rArr(SinA//2cosB//2-sinB//2cosA//2)/(cosA//2,cosB//2)`
`=(sin B//2cosC//2-sinC//2.cosB//2)/(cosB//2.cosC//2)`
`rArr(sin((A-B)/(2)))/(cosA//2)=(sin((B-C)/(2)))/(cosC//2)`
`rArr sin((A-B)/(2))sin((A+B)/(2))=sin((B-C)/(2)s)in((B+C)/(2))`
`rArr cos B-cosA=cosC-cosB`
Hence, `cosA,cosB,cosC` are in AP.