Answer to Question #301960 in Trigonometry for Nabs

Question

Show the derivation of the formula.

Cos u/2=±√1+cos u/2.

Answer 1

cos(u+v) = cos u cos v - sin u sin v

Put v = u, we get

cos (u + u) = cos u cos u - sin u sin u

cos 2u = $cos^2u - sin^2 u$

$cos 2u = cos^2u - (1-cos^2u)$

$cos 2u = 2 cos^2 u - 1$

On rearranging, we get,

$2 cos^2 u = 1 +cos 2u$

$cos ^2u = \frac{1+cos 2u}{2}$

$cos u =\sqrt{(\frac{1+cos2u}{2})}$

put u = u/2

$cos\frac{u}{2} = \sqrt{(\frac{(1+cos u)}{2})}$