If $$0 < \phi < \frac{\pi }{2},x = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\phi } ,y = \sum\limits_{n = 0}^\infty {{{\sin }^{2n}}\phi } $$         and $$z = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\phi \cdot {{\sin }^{2n}}\phi } $$     then

Solution :
$$\eqalign{ & x = 1 + {\cos ^2}\phi + {\cos ^4}\phi + .....\,{\text{to }}\infty = \frac{1}{{1 - {{\cos }^2}\phi }} = \frac{1}{{{{\sin }^2}\phi }}. \cr & {\text{Similarly, }}y = \frac{1}{{1 - {{\sin }^2}\phi }} = \frac{1}{{{{\cos }^2}\phi }},z = \frac{1}{{1 - {{\cos }^2}\phi \cdot {{\sin }^2}\phi }} \cr & \therefore \,\,xyz = \frac{1}{{{{\sin }^2}\phi \cdot {{\cos }^2}\phi \left( {1 - {{\sin }^2}\phi \,{{\cos }^2}\phi } \right)}} \cr & xyz = \frac{{\left( {1 - {{\sin }^2}\phi \cdot {{\cos }^2}\phi } \right) + {{\sin }^2}\phi \cdot {{\cos }^2}\phi }}{{{{\sin }^2}\phi \cdot {{\cos }^2}\phi \left( {1 - {{\sin }^2}\phi \,{{\cos }^2}\phi } \right)}} \cr & xyz = \frac{1}{{{{\sin }^2}\phi \,{{\cos }^2}\phi }} + \frac{1}{{1 - {{\sin }^2}\phi \,{{\cos }^2}\phi }} = xy + z. \cr} $$