Question

Verify ((csc^2x) + (2cscx) -3) / ((csc^2x) - 1) = (cscx+3) / (cscx + 1)

Answer 1

`(cosec^2\ x + 2cosec\ x - 3)/(cosec^2x - 1) = (cosec\ x + 3)/(cosec\ x + 1)`

Solution:

Given that, we have to verify

`(cosec^2\ x + 2cosec\ x - 3)/(cosec^2x - 1) = (cosec\ x + 3)/(cosec\ x + 1)`

Take the LHS

`(cosec^2\ x + 2cosec\ x - 3)/(cosec^2x - 1)`

Use the following identity, for denominator

`a^2 - b^2 = (a+b)(a-b)`

Therefore,

`(cosec^2\ x + 2cosec\ x - 3)/((cosecx + 1)(cosec\ x - 1))`

Numerator can be rewritten as:

`((cosec\ x +3)(cosec\ x - 1))/((cosecx + 1)(cosec\ x - 1))`

Cancel the common terms

`((cosec\ x +3))/((cosecx + 1))`

Thus, LHS = RHS

Thus proved