Question

How do you prove `cscx /(1 + cscx) + cscx /(1 - cscx) = (- 2sinx) / cos^2x`?

Answer 1

I would start as:

`(cscx(1-cscx)+cscx(1+cscx))/((1+cscx)(1-cscx))=-2sinx/cos^2x`
`(cscxcancel(-csc^2x)+cscxcancel(+csc^2x))/(1-csc^2x)=-2sinx/cos^2x`
`(2csc(x))/(1-csc^2x)=-2sinx/cos^2x`

I would use the fact that:
`cscx=1/sinx`
So:
`cancel(2)/cancel(sinx)(cancel(sin)^cancel(2)x)/(sin^2x-1)=-cancel(2)cancel(sinx)/cos^2x` collect `-1` on the left side:
`-1/(1-sin^2x)=-1/cos^2x`
But `1-sin^2x=cos^2x`