Question

Prove the following trigonometric identities.

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

Answer 1

We need to prove  `(cosec A)/(cosec A  - 1) + (cosec A)/(cosec A = 1) = 2 sec^2 A`

Using identity `a^2 - b^2 = (a + b)(a - b)` we get

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

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

Further, using the property  `1 + cot62 theta = cosec^2 theta` we get

So

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

`= (2cosec^2 A)/cot^2 A`

`= (2)(1/sin^2 A)((cos^2 A)/(sin^2 A))`

`= 2(1/cos^2 A)`

`= 2 sec^2 A`

Hence proved.