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

1 Answer
Jul 25, 2017

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.