Answer:
Given expression is [tex]\frac{1}{\text{cosec}A-1}-\frac{1}{\text{cosecA+1}}[/tex] = 2tan²A
We have to prove this identity.
[tex]\frac{1}{\text{cosec}A-1}-\frac{1}{\text{cosecA+1}}=\frac{(\text{cosecA+1})-(\text{cosecA-1})}{(\text{cosecA-1})(\text{cosecA+1})}[/tex]
= [tex]\frac{(\text{cosecA}-\text{cosecA})+(1+1)}{cosec^2A-1}[/tex]
= [tex]\frac{2}{cosec^{2}A-1}[/tex]
= [tex]\frac{2}{cot^{2}A}[/tex] [From the identity, (cosec²A - 1) = cot²A]
= 2tan²A [From the identity, cotA = [tex]\frac{1}{\text{tan}A}[/tex]]
Hence the given equation is proved.
