If x,y,z are in A.P and tan-1x,tan-1yandtan-1z are also in A.P., then
x=y=z
x=y=-z
x=1,y=2,z=3
x=2,y=4,z=6
Explanation for the correct option.
x,y,z are in A.P, so
2y=x+z....1
tan-1x,tan-1yandtan-1z are in A.P, so
2tan-1y=tan-1x+tan-1z⇒tan-12y1-y2=tan-1x+z1-xzby2tan-1θ=tan-12θ1-θ2andtan-1A+tan-1B=tan-1A+B1-AB⇒2y1-y2=2y1-xzfrom1⇒1-xz=1-y2⇒y2=xz
This shows that x,y,z are in A.P as well as in G.P.
Therefore, x=y=z
Hence, option A is correct.