If x, y, z are in A.P. and `tan^(-1) x, tan^(-1) y and tan^(-1)z` are also in A.P. then show that x=y=z and y≠0

Options

A. x=y=z
B. xy=yz
C. `x^(2)=yz`
D. `z^(2)=xy`

Answer

A

Explanation

We have,
x,y,z are in A.P.
`rArr" "2y=x+z` . . .(i)
Also, `tan^(-1)xtan^(-1)z` are in A.P.
`rArr" "2tan^(-1)y=tan^(-1)x+tan^(-1)z`
`rArr" "tan^(-1)((2y)/(1-y^(2)))=tan^(-1)((x+z)/(1-xz))`
`rArr" "(2y)/(1-y^(2))=(x+z)/(1-xz)`
`rArr" "1-y^(2)=1-xz" [Using (i)]"`
`rArr" "y^(2)=xz`
`rArr" "` x,y,z are in G.P. . . . (ii)
From (i) and (ii), we get,
x=y=z