Question

Suppose x, y, z are in A.P. and that sin (y + z) = x, sin (z + x) = y and sin (x + y) = z. Prove that tan x, tan y, tan

z are in A.P. (assume that sin (2y) ≠ 0).​

Answer 1

Answer:

a,b,c are in AP = = a-b=b-c. sin (-x+y+z) - sin (x-y+z) = sin (x-y+z) -sin (x+y-z) . 2sin (( -2x+2y)/2) cos ((2z)/2) =2sin ((-2y+2z)/2) cos ((2x)/2). sin (y-x) cos (z) =sin (z-y) cos (x). cosz(sinycosx -cosxsiny) (sinzcosy-cosysinz) cosx. 2coszsinycosx = sinzcosycosx + sinxcoszcosy. divide by cosxcosycosz: 2tany= t anz + tanx == tanx,tany, tanz are in AP.

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