Let $$0 < x < {\pi \over 4}$$ then $$\left( {\sec 2x - \tan 2x} \right)$$ equals

Let $$n$$ be a positive integer such that $$\sin {\pi \over {2n}} + \cos {\pi \over {2n}} = {{\sqrt n } \over 2}.$$ Then

If $$\omega \,$$ is an imaginary cube root of unity then the value of $$\sin \left\{ {\left( {{\omega ^{10}} + {\omega ^{23}}} \right)\pi - {\pi \over 4}} \right\}$$ is

Suppose Z₁, Z₂, Z₃ are the vertices of an equilateral triangle inscribed in the circle $$\left| Z \right| = 2.$$ If Z₁ = $$1 + i\sqrt 3 $$ then Z2 = ......., Z₃ =..............