Question

If sin(y + z – x), sin(z + x – y), sin(x + y – z) are in A.P, then prove that tan x, tan y and tan z are in A.P.

Answer 1

In A.P. commom difference are equal, namely t₂ - t₁ = t₃ - t₂

sin(z + x – y) – sin(y + z – x) = sin(x + y – z) – sin(z + x – y)

cos z sin (x – y) = cos x sin (y – z)

cos z (sin x cos y – cos x sin y) = cos x (sin y cos z – cos y sin z)

Divide bothsides by cos x cos y cos z we get

tan x – tan y = tan y – tan z

Multiply both sides by (-1) we get,

tan y – tan x = tan z – tan y

This means tan x, tan y, and tan z are in A.P.

Hence proved.