jee-advanced 2003 Q18

jee-advanced · India · mains Proof Direct Proof of a Stated Identity or Equality
If In is the area of $n$ sided regular polygon inscribed in a circle of unit radius and On be the area of the polygon circumscribing the given circle, prove that $$I _ { n } = \frac { O _ { n } } { 2 } \left( 1 + \sqrt { 1 - \left( \frac { 2 l _ { n } } { n } \right) ^ { 2 } } \right)$$
If $z$ is a complex number satisfying $| z | = 1$ and $z \neq - 1$, then the real part of $w = \frac { z - 1 } { z + 1 }$ is
If In is the area of $n$ sided regular polygon inscribed in a circle of unit radius and On be the area of the polygon circumscribing the given circle, prove that $$I _ { n } = \frac { O _ { n } } { 2 } \left( 1 + \sqrt { 1 - \left( \frac { 2 l _ { n } } { n } \right) ^ { 2 } } \right)$$