Simplify \(\large \dfrac { \csc { \theta } }{ \sin { \theta } } - \dfrac { \cot { \theta } }{ \tan { \theta } } = \, ? \)
csc(x) = 1 / sin(x)
---> csc(x) / sin(x) = [ 1 / sin(x) ] / sin(x) = 1 / sin2(x)
cot(x) = cos(x) / sin(x)
tan(x) = sin(x) / cos(x)
---> cot(x) / tan(x) = [ cos(x) / sin(x) ] / [ sin(x) / cos(x) ] = cos2(x) / sin2(x)
So, we now have: [ 1 / sin2(x) ] - [ cos2(x) / sin2(x) ]
---> [ 1 - cos2(x) ] / sin2(x)
---> sin2(x) / sin2(x) = 1