Question

How do you solve `\frac { 1} { \sin y - 1} - \frac { 1} { \sin y + 1} = - \sec y`?

Answer 1

No solution

Explanation:

Combine the left side using LCD = `(sin y)^2-1` i.e.

`1/(siny-1)-1/(siny+1)=(siny+1-(siny-1))/(sin^2y-1)`

or `2/ [(sin^2y-1)`

or `2/-(1-sin^2y)=-2/cos^2y`

The right side is `-1/cosy`

Cross multiply to get

`-2/cos^2y=-1/cosy`

or `-2cosy=-cos^2y`

`cos^2y-2cosy=0`

or `cosy(cosy-2)=0`

but as `cosy-2!=0`

Hence `cosy=0`

i.e. `y=(npi)/2, (3npi)/2`

or in general `y=(2n+1)pi/2`, where `n` is an integer.

But for all these `siny-1` or `siny+1` is undefined,

Hence, no solution.