Let $a _ { 0 } = \frac { 1 } { 2 }$ and $a _ { n }$ be defined inductively by
$$a _ { n } = \sqrt { \frac { 1 + a _ { n - 1 } } { 2 } } , n \geq 1 .$$
(a) Show that for $n = 0,1,2 , \ldots$,
$$a _ { n } = \cos \theta _ { n } \text { for some } 0 < \theta _ { n } < \frac { \pi } { 2 } ,$$
and determine $\theta _ { n }$.
(b) Using (a) or otherwise, calculate
$$\lim _ { n \rightarrow \infty } 4 ^ { n } \left( 1 - a _ { n } \right)$$