solution:-(Method 1)
LHS= 1 /(cosecθ-cotθ) - 1/ sinθ
=(cosec^2θ-cot^2θ) /(cosecθ-cotθ) -cosecθ
=(cosecθ-cotθ) (cosecθ+cotθ) /(cosecθ-cotθ) -cosecθ
= cosecθ + cotθ - cosecθ
=cotθ
RHS= 1/ sinθ - 1/(cosecθ+cotθ)
=cosecθ-(cosec^2θ-cot^2θ) /(cosecθ+cotθ)
=cosecθ - (cosecθ+cotθ) (cosecθ-cotθ) /(cosecθ+cotθ)
= cosecθ -(cosecθ -cotθ)
=cosecθ - cosecθ +cotθ
=cotθ
LHS=RHS proved
(Method 2)
LHS= 1 /(cosecθ-cotθ) - 1/ sinθ
=(cosec^2θ-cot^2θ) /(cosecθ-cotθ) -cosecθ
=(cosecθ-cotθ) (cosecθ+cotθ) /(cosecθ-cotθ) -cosecθ
= cosecθ + cotθ - cosecθ
=cosecθ - cosecθ + cotθ
= cosecθ - (cosecθ -cotθ)/1
=1/ sinθ - (cosecθ -cotθ)/(cosec^2θ-cot^2θ)
=1/ sinθ - (cosecθ-cotθ) /(cosecθ-cotθ)(cosecθ+cotθ)
= 1/ sinθ - 1/(cosecθ+cotθ)=RHS proved
same as above
1/cosec a-cot a-1/sin a=1/sin a-1/cosec a+cot a
prove that (1/coseca-cota)-(1/sina)=(1/sina)-(1/coseca+cota)
1 by cosec A minus cot A minus 1 by sin A
1 by cosec theta minus cot theta
one by cosec theta is equal to sin theta
1 upon cosec theta minus cot theta
prove that one upon cosec theta minus cot theta is equal to cosec theta + cot theta
