If x = cosecθ - sinθ and y = secθ - cosθ, then the relation between x and y is?

Answer: Option B

$$\begin{array}{rl}& x=\text{cosec}\theta -\sin \theta \\ & y=\sec \theta -\cos \theta \\ & \text{Put }\theta =45^{\circ }\\ & x=\sqrt{2}-\frac{1}{\sqrt{2}}=\frac{1}{\sqrt{2}}\\ & y=\sqrt{2}-\frac{1}{\sqrt{2}}=\frac{1}{\sqrt{2}}\\ & \text{by options (B) }{x}^2{y}^2\left(x^2+y^2+3\right)\\ & ={\left(\frac{1}{\sqrt{2}}\right)}^2\times {\left(\frac{1}{\sqrt{2}}\right)}^2\\ & \left[{\left(\frac{1}{\sqrt{2}}\right)}^2\times {\left(\frac{1}{\sqrt{2}}\right)}^2+3\right]\\ & =\frac{1}{2}\times \frac{1}{2}\left(\frac{1}{2}+\frac{1}{2}+3\right)\\ & =\frac{1}{4}\left(1+3\right)\\ & =1\left(\text{satisfy}\right)\end{array}$$