If x=secϕ−tanϕ and y=cosecϕ+cotϕ then
x=y+1y−1
x=y−1y+1
y=1+x1−x
xy+x−y+1=0
We have x=1−sinϕcosϕ,y=1+cosϕsinϕ
Multiplying, we get
xy=(1−sinϕ)(1+cosϕ)cosϕsinϕ
⇒ xy+1=1−sinϕ+cosϕ−sinϕcosϕ+sinϕcosϕcosϕsinϕ=1−sinϕ+cosϕcosϕsinϕ and x−y=(1−sinϕ)sinϕ−cosϕ(1+cosϕ)cosϕsinϕ=sinϕ−sin2ϕ−cosϕ−cos2ϕcosϕsinϕ=sinϕ−cosϕ−1cosϕsinϕ=−(xy+1)
thus,xy+x−y+1=0,x=y−1y+1, and y=1+x1−x.