Question

Prove- sin theta/cot theta + cosec theta=2+sin theta/cot theta- cosec theta

Answer 1

LHS
sin α/(cotα + cosecα) =
=sin α(cosecα-cotα)/(cotα + cosecα)(cosecα-cotα) .............multiply both denominator and numerator by (cosecα-cotα)
=1-cosα................................... (since cosecα=1/sinα & cosec^2α-cot^2α=1)

RHS
2+sin α/(cot α- cosecα)
= 2 + sin α(cot α+cosecα) /(cot α- cosecα) (cot α+cosecα) ..........multiply both denominator and numerator by (cosecα+cotα)
=2 + (cosα + 1)/-1 .... .................... (since cosecα=1/sinα & cosec^2α-cot^2α=1)
=2-cosα-1
=1-cosα.....


hence LHS = RHS