Trig Help Please?

Question

I cannot figure this problem out.
You can only make changes to one side.
Prove these 2 are equal:

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

I can't figure this out.

Answer 1

sin(x) / (1+cos(x)) = (1- cos(x)) / sin(x)
RHS
(1- cos(x)) / sin(x) = (1- cos(x)) / sin(x) * (1+ cos(x)) / (1+cosx)
= 1- cos^2x/sinx(1+cosx)
= sin^2x/sinx(1+cosx)
= sin x/ (1+cosx) = LHS

Answer 2

Why that restriction on "only mak[ing] changes to one side"? It seems extremely artificial.

A. The obvious way to prove this identity is to start from the well-known identity:

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

Now, just divide both sides of the latter equality by the product
[sin(x) (1 + cos(x))].

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

B. If you INSIST on only "changing one side," the method involves reversing part of what we just did. Start from the left-hand side (LHS), and multiply by a cunningly expressed fraction that is in fact identically one (i.e. 1), that is [(1 - cos(x)) / (1 - cos(x))]. Then :

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

= sin(x) (1 - cos(x)) / [1 - cos^2 (x))] = sin(x) (1 - cos(x)) / sin^2(x)

= (1- cos(x)) / sin(x). This is the RHS. QED.

Live long and prosper.

Answer 3

Take the LHS ie
sinx/1+cosx
Now multiply the numerator and denominator by 1-cosx
so it will become sinx(1-cosx)/(1-cos^2x)
=sinx(1-cosx)/sin^2x
hence 1-cosx/sinx
hence proved