Question #13d24

1 Answer
Nov 17, 2016

see below

Explanation:

sqrt((1+cosA)/2)=cos (A/2 )1+cosA2=cos(A2)

We will prove this by using the double argument formula for cosine.

Recall that one of the formula for cos 2Acos2A is cos2A=2cos^2A-1cos2A=2cos2A1

So if we isolate cos A then we have

(cos 2A+1)/2=cos^2 Acos2A+12=cos2A

+-sqrt ( (1+cos2A)/2)=cos A±1+cos2A2=cosA

So from this formula we can see that you double angle A inside the square root. Since A in this case is 1/2A12A then we have 2A=2(1/2A)=A2A=2(12A)=A,

+-sqrt ( (1+cos2(1/2A))/2)=cos (1/2A)±1+cos2(12A)2=cos(12A)

:. +-sqrt((1+cosA)/2)=cos (A/2 )