If sinα .sec(30° + α) = 1, (0 < α < 60°), then the value of sinα + cos2α is?

Solution(By Examveda Team)

$$\eqalign{ & {\text{sin }}\alpha .\sec \left( {{{30}^ \circ } + \alpha } \right) = 1 \cr & {\bf{Shortcut\,\, method:}} \cr & {\text{Put }}\alpha {\text{ value between }}{{\text{0}}^ \circ }{\text{ to 6}}{{\text{0}}^ \circ } \cr & {\text{If }}\alpha = {30^ \circ } \cr & \Rightarrow {\text{sin 3}}{0^ \circ }.\sec \left( {{{30}^ \circ } + {{30}^ \circ }} \right) = 1 \cr & \Rightarrow {\text{sin 3}}{0^ \circ }.\sec {60^ \circ } = 1 \cr & \Rightarrow \frac{1}{2} \times 2 \cr & \Rightarrow 1 \cr & 1 = 1\left( {{\text{Satisfy}}} \right) \cr & {\text{So}},\alpha = {30^ \circ } \cr & = \sin \alpha + \cos 2\alpha \cr & = \sin {30^ \circ } + \cos 2 \times {30^ \circ } \cr & = \sin {30^ \circ } + \cos {60^ \circ } \cr & = \frac{1}{2} + \frac{1}{2} \cr & = 1 \cr } $$