Q- Prove :-

1) cosec theta ( sec theta - 1) - cot theta ( 1- cos theta) = tan theta - sin theta .

2) 1- sinA cosA / cosA ( secA - cosecA ) . sin²A - cos²A / sin³A + cos³A = sin A

3) cos theta / 1- sin theta = 1+cos theta + sin theta / 1+ cos theta - sin theta

1) let theta = x (just for ease of writing)

LHS=

cosec x ( sec x-1 )-cot x ( 1-cos x )

= 1 / sin x cosx - 1/sin x - cos x / sin x - (- cos^2 x / sin x )

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

= sin^2 x/ cos x sin x - sin^2 x / sin x

= sin x / cos x-sin x= tan x - sin x