We have $x=\sec \phi-\tan \phi$
and $y=\operatorname{cosec} \phi+\cot \phi$
Now, $x y=(\sec \phi-\tan \phi)(\operatorname{cosec} \phi+\cot \phi)$
$\Rightarrow x y=\sec \phi \cdot \operatorname{cosec} \phi-\operatorname{cosec} \phi \cdot \tan \phi+$
$\sec \phi \cdot \cot \phi-\tan \phi \cdot \cot \phi$
$\Rightarrow x y=\sec \phi \cdot \operatorname{cosec} \phi-\frac{1}{\cos \phi}+\frac{1}{\sin \phi}-1$
$\Rightarrow 1+x y=\sec \phi \operatorname{cosec} \phi-\sec \phi+\operatorname{cosec} \phi$
From Eqns. (i) and (ii), we get
$x-y=\sec \phi-\tan \phi-\operatorname{cosec} \phi-\cot \phi$


$$
\begin{aligned}
&\Rightarrow x-y=\sec \phi-\operatorname{cosec} \phi-\frac{\sin \phi}{\cos \phi}-\frac{\cos \phi}{\sin \phi} \\
&\Rightarrow x-y=\sec \phi-\operatorname{cosec} \phi-\left(\frac{\sin ^2 \phi+\cos ^2 \phi}{\sin \phi \cdot \cos \phi}\right) \\
&\Rightarrow x-y=\sec \phi-\operatorname{cosec} \phi-\frac{1}{\sin \phi \cdot \cos \phi} \\
&\Rightarrow x-y=\sec \phi-\operatorname{cosec} \phi-\operatorname{cosec} \phi \cdot \sec \phi \\
&\Rightarrow x-y=-(\sec \phi \cdot \operatorname{cosec} \phi-\sec \phi+\operatorname{cosec} \phi) \\
&\Rightarrow x-y=-(x y+1) \quad[\text { from Eqn. (iii)] } \\
&\Rightarrow x y+x-y+1=0 \quad \text { Henceproved. }
\end{aligned}
$$