trigonometric functions?

Question

tanxsinx over tanx+sinx = tan-sinx over tanxsinx

Answer 1

tanx sinx / (tanx + sinx)

[sinx/cosx (sinx)]/[sinx/cosx + sinx/1]

[sin^2x /cosx]/[sinx/cosx + sinxcosx/cosx]

(sin^2x / cosx)/[sinx(1-cosx)/cosx]

sin^2 x / [sinx(1 - cosx)]

(1 - cos^2 x)/[sinx(1 - cosx)]

[(1 + cosx)(1 - cosx)]/[sin x (1 - cosx)]

(1 + cosx) / sinx

tanx (1 + cosx) / (tan x sin x)

(tanx+ tanxcosx) / (tanx sinx)

(tanx + (sinx/cosx)(cosx)) / tanx sinx

(tanx + sinx)/(tanx sinx)

Answer 2

(tan(x)sin(x))/(tan(x) + sin(x)) = (tan(x) - sin(x))/(tan(x)sin(x))

((sin(x)/cos(x))sin(x))/((sin(x)/cos(x)) + sin(x)) = ((sin(x)/cos(x)) - sin(x))/((sin(x)/cos(x))sin(x))

((sin(x)^2)/cos(x))/((sin(x) + sin(x)cos(x))/(cos(x))) = ((sin(x) - sin(x)cos(x))/(cos(x))/((sin(x)^2)/(cos(x)))

((sin(x)^2)/(cos(x))*((cos(x))/(sin(x) + sin(x)cos(x))) = ((sin(x) - sin(x)cos(x))/(cos(x)) * (cos(x)/sin(x)^2)

((cos(x)sin(x)^2)/((cos(x))(sin(x) + sin(x)cos(x))) = ((cos(x))(sin(x) - sin(x)cos(x))/(cos(x)sin(x)^2)

((sin(x)^2)/((sin(x))(1 + cos(x))) = ((sin(x))(1 - cos(x)))/(sin(x)^2)

(sin(x))/(1 + cos(x)) = (1 - cos(x))/(sin(x))

Cross multiply

sin(x)^2 = (1 - cos(x))(1 + cos(x))
sin(x)^2 = 1 + cos(x) - cos(x) - cos(x)^2
sin(x)^2 = 1 - cos(x)^2
sin(x)^2 = sin(x)^2