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

Solution (By Examveda Team)

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