Question

If sin(z + y - x), sin(x + z-y), sin(y + x - Z) be in A.P., prove that tanx, tany, tanz are also in A.P.

JEE/Engineering Exams

Maths

Solution

326

5.0 (2 ratings)

( sin (3+y-x), sin (x+3-y), sin (x+y-3): A P ) ( Rightarrow quad delta u_{1}(x+z-y)-dot{sin }(g+y-x)=sin (x+y-z)-sin (x+z y) ) ( Rightarrow 2 sin left(frac{2 x-2 y}{2}right) cos left(frac{2 z}{2}right)=2 sin left(frac{2 y-2 z}{2}right) cos left(frac{2 x}{2}right) ) ( Rightarrow quad sin (x-y) cos (3)=sin (y-3) cos (x) ) ( Rightarrow quad sin x cos y-cos x sin y=operatorname{sen} y cos z-sin z cos y ) ( cos x ) ( theta 32 ) ( Rightarrow tan x cos y-cos x sin y=sin y-tan 3 cos y ) ( Rightarrow quad(tan x+tan z) cos y=2 sin y ) ( Rightarrow tan x+tan z=2 tan y ) ( Rightarrow ) tanx, truiy, tang are in AP.