Question

cot x+ cosec x-1/cot x- cosec x+1 = 1+cos x/sin x.
prove this

Answer 1

LHS=(cotx+cosecx)-1/(cotx-cosecx+1)
=(cosecx+cotx)-(cosec²x-cot²x)/(cotx-cosecx+1)
=(cosecx+cotx)[1-(cosecx-cotx)]/(cotx-cosecx+1)
=(cosecx+cotx)(cotx-cosecx+1)/(cotx-cosecx+1)
=(cosecx+cotx)
=(1/sinx+cosx/sinx)
=1+cosx/sinx RHS

Answer 2

Given

(cot x+ cosec x-1)/(cot x- cosec x+1)
→(cot x+ cosec x-1)/(cot x- cosec x+cosec²x -cot²x) ............as we know that cosec²x -cot²x = 1
so putting 1 = cosec²x -cot²x
→(cot x+cosec x-(cosec²x -cot²x))/(cot x- cosec x+1)

see picture!
in 3rd line take cosecx +cotx common in both terms....
hope it helped you!!!