If x=sec⁡ϕ−tan⁡ϕ and y=cosec⁡ϕ+cot⁡ϕ then

We have   x=1−sin⁡ϕcos⁡ϕ,y=1+cos⁡ϕsin⁡ϕMultiplying, we get xy=(1−sin⁡ϕ)(1+cos⁡ϕ)cos⁡ϕsin⁡ϕ⇒ xy+1=1−sin⁡ϕ+cos⁡ϕ−sin⁡ϕcos⁡ϕ+sin⁡ϕcos⁡ϕcos⁡ϕsin⁡ϕ=1−sin⁡ϕ+cos⁡ϕcos⁡ϕsin⁡ϕ and  x−y=(1−sin⁡ϕ)sin⁡ϕ−cos⁡ϕ(1+cos⁡ϕ)cos⁡ϕsin⁡ϕ=sin⁡ϕ−sin2⁡ϕ−cos⁡ϕ−cos2⁡ϕcos⁡ϕsin⁡ϕ=sin⁡ϕ−cos⁡ϕ−1cos⁡ϕsin⁡ϕ=−(xy+1)thus,xy+x−y+1=0,x=y−1y+1, and y=1+x1−x.