If $$\frac{{{{\sin }^2}\theta - 3\sin \theta + 2}}{{{{\cos }^2}\theta }} = 1,$$     where 0° < θ < 90°, then what is the value of (cos2θ + sin3θ + cosec2θ)?

Solution (By Examveda Team)

$$\eqalign{ & \frac{{{{\sin }^2}\theta - 3\sin \theta + 2}}{{{{\cos }^2}\theta }} = 1 \cr & {\text{Put }}\theta = {30^ \circ } \cr & \frac{{{{\sin }^2}{{30}^ \circ } - 3\sin {{30}^ \circ } + 2}}{{{{\cos }^2}{{30}^ \circ }}} = 1 \cr & \frac{1}{4} - \frac{3}{2} + 2 = \frac{3}{4} \cr & \frac{3}{4} = \frac{3}{4} \cr & \cos 2\theta + \sin 3\theta + {\text{cosec}}\,2\theta \cr & = \cos {60^ \circ } + \sin {90^ \circ } + {\text{cosec}}\,{60^ \circ } \cr & = 1 + \frac{1}{2} + \frac{2}{{\sqrt 3 }} \cr & = \frac{{9 + 4\sqrt 3 }}{6} \cr} $$