If `a^(2),b^(2)` and `c^(2)` are in AP, then cotA, cotB and cotC are in

Options

A. AP
B. GP
C. HP
D. AGP

Answer

A

Explanation

`cotA=(cosA)/(sinA)=(b^(2)+c^(2)-a^(2))/(2bcsinA)=(b^(2)+c^(2)-a^(2))/(4Delta)`
Similarly, `cotB=(a^(2)+c^(2)-b^(2))/(4Delta)`
and `cotC=(a^(2)+b^(2)-c^(2))/(4Delta)`
Given that `a^(2),b^(2)` and `c^(2)` are in AP.
`implies -2a^(2),-2b^(2),-2c^(2)` are in AP.
`implies (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 AP.
`implies b^(2)+c^(2)-a^(2),c^(2)+a^(2)-b^(2),a^(2)+b^(2)-c^(2)` are in AP.
`implies(b^(2)+c^(2)-a^(2))/(4Delta),(c^(2)+a^(2)-b^(2))/(4Delta),(a^(2)+b^(2)-c^(2))/(4Delta)` are in AP.
`impliescotA,cotB,cotC` are in AP.