Question

1 / cosec A -1 - 1 / cosec A + 1​

Answer 1

Given expression is
`\frac{1}{\text{cosec}A-1}-\frac{1}{\text{cosecA+1}}` = 2tan²A

We have to prove this identity.

`\frac{1}{\text{cosec}A-1}-\frac{1}{\text{cosecA+1}}=\frac{(\text{cosecA+1})-(\text{cosecA-1})}{(\text{cosecA-1})(\text{cosecA+1})}`

=
`\frac{(\text{cosecA}-\text{cosecA})+(1+1)}{cosec^2A-1}`

=
`(2)/(cosec^(2)A-1)`

=
`(2)/(cot^(2)A)` [From the identity, (cosec²A - 1) = cot²A]

= 2tan²A [From the identity, cotA =
`\frac{1}{\text{tan}A}`]

Hence the given equation is proved.