If 2cos²θ + 3sinθ = 3, where 0° < θ < 90°, then what is the value of sin²2θ + cos²θ + tan²2θ + cosec²2θ?

Solution (By Examveda Team)

$$\eqalign{ & 2{\cos ^2}\theta + 3\sin \theta = 3 \cr & {\text{Let }}\theta = {30^ \circ } \cr & 2{\cos ^2}{30^ \circ } + 3\sin {30^ \circ } = 3 \cr & 2 \times \frac{3}{4} + 3 \times \frac{1}{2} = 3 \cr & 3 = 3 \cr & {\sin ^2}2\theta + {\cos ^2}\theta + {\tan ^2}2\theta + {\text{cose}}{{\text{c}}^2}2\theta \cr & = {\sin ^2}{60^ \circ } + {\cos ^2}{30^ \circ } + {\tan ^2}{60^ \circ } + {\text{cose}}{{\text{c}}^2}{60^ \circ } \cr & = \frac{3}{4} + \frac{3}{4} + 3 + \frac{4}{3} \cr & = \frac{3}{2} + 3 + \frac{4}{3} \cr & = \frac{{9 + 18 + 8}}{6} \cr & = \frac{{35}}{6} \cr} $$