(Sin x Tan x) / (Tan x - Sin x) = (Tan x + Sin x) / (Sin x Tan x)

Question

(Sin x Tan x) / (Tan x - Sin x) ≡ (Tan x + Sin x) / (Sin x Tan x)

 

LHS, (Sin x Tan x) / (Tan x - Sin x)

Answer 1

(sinxtanx)/(tanx-sinx)=(tanx+sinx)/(sinxtanx)

working with the left side first...

[(sinx)(sinx/cosx)]/(tanx-sinx)

(sin²x/cosx)/[(sinx/cosx)-sinx]

(sin²x/cosx)/([sinx-sinxcosx]/cosx)

(sin²x/cosx)*(cosx/(sinx-sinxcosx)

sin²x/(sinx-sinxcosx)

sinx/(1-cosx)

working with the right side now...

[(sinx/cosx)+sinx)]/[(sinx)(sinx/cosx)]

[(sinx+sinxcosx)/cosx]/(sin²x/cosx)

[(sinx+sinxcosx)/cosx]*(cosx/sin²x)

(sinx+sinxcosx)/sin²x

(1+cosx)/sinx

equate the two sides...

sinx/(1-cosx)=(1+cosx)/sinx

cross multiply...

sin²x=1-cos²x

sin²x+cos²x=1 which is a trig identity