Prove that (sin(2x) - sin(x)) / (cos(2x) - cos(x) + 1) is equal to Tan(x)?

Question

Using trigonometric identities

Answer 1

O.K., first work separately on numerator and denominator.

sin 2x = 2 sin x cos x (an identity); so:

sin 2x - sin x = (2 sin x) cos x - sinx = sin x (2 cos x -1).

cos 2x = 2 cos^2 (x) - 1 (an identity); so:

cos 2x - cos x + 1 = [2 cos^2 (x) - 1] - cos x + 1 = cos x (2 cos x -1)

Note the common factor (2 cos x -1); it divides out, so:

[sin 2x - sin x] / [cos 2x - cos x + 1] = sin x / cos x = tan x.

Live long and prosper.

P.S. It's so REFRESHING to see the proper use of brackets in posing the question. THANK YOU!

Answer 2

Remember sinx=2sinx cosx
cos2x=cos^2 x- sin^2 x

numerator= 2sinxcosx-sinx
=sinx(2cosx-1)

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

Hence

sinx(2cosx-1)
--------------------
cosx ( 2cosx-1)

=sinx/cosx
=tanx

Answer 3

Dang, that looks harder than Chinese arithmetic! Everyone knows Chinese arithmetic is really really hard.

Answer 4

(sin(2X)-sin(X))/(cos(2X)-cos(X)+1)

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

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

=(sin(X)(2cos(X)-1))/
(cos(X)(2cos(X)-1))

=tan(X)