Simplify: (Sin x/1 - cos x) + (sin x/1 + cos x) = 2 csc x

I have:
(sin x/1 – cos x)(1 + cos x/1 + cos x) + (sin x/1 + cos x)(1 – cos x/1 – cos x) = 2csc x
(Sin x + sin cos x)/(1-cos^2) + (sin x – sin cos x)/(1-cos^2)= 2 csc x
(Sin x + sin cos x)/(sin^2) + (sin x – sin cos x)/(sin^2)= 2 csc x
2 sin/sin^2 = 2 csc x
Where did I go wrong? It's supposed to equal itself.

Question

Simplify: (Sin x/1 - cos x) + (sin x/1 + cos x) = 2 csc x 

I have:
(sin x/1 – cos x)(1 + cos x/1 + cos x) + (sin x/1 + cos x)(1 – cos x/1 – cos x) = 2csc x
(Sin x + sin cos x)/(1-cos^2) + (sin x – sin cos x)/(1-cos^2)= 2 csc x
(Sin x + sin cos x)/(sin^2) + (sin x – sin cos x)/(sin^2)= 2 csc x
2 sin/sin^2 = 2 csc x
Where did I go wrong? It's supposed to equal itself.

anonymous · Answer

looks to me like you didn't do anything wrong, though you did it a different way than I did.$$\frac{ 2sinx }{ (sinx)^2 } = \frac{ 2 }{ sinx } = 2cscx$$
does that look right?

anonymous · Answer

Oh I just stopped because it looked wrong. Thank you very much!

anonymous · Answer

good work, those can be tricky