Sin x/[1-cos(-x)] = csc x + cot x

Please Verify the Identity.

I have been going in circle with this one... Just a hint would be nice. Thank you.

2 answers

LS = sinx/(1 - cos(-x))
first of all cos(-x) = cosx, so let's change that

= sinx/(1 - cosx)
now multiply top and bottom by (1+cosx)/(1+cos),
thus changing the appearance, but not its value

= sinx/(1-cosx) * (1+cosx)/(1+cos)
= sinx(1+cosx)/(1 - cos^2 x)
= (sinx + sinxcosx)/sin^2 x
= sinx/sin^2 x + sinxcosx/sin^2 x
= 1/sinx + cosx/sinx
= cscx + cotx
= RS
Thank you so much!