Trigonometric sequence - Find Un out of Un+1 - hard

[JPC]

Homework Statement

Un a sequence

0 < x < pi/2

U0 = 2cosx

Un+1 = quroot( 2 + Un)

1) Calculate U1, U2, and U3 in function of x (simplify maximum)

2) Show that Un = 2cos( x / 2^n)

Homework Equations

The Attempt at a Solution

1)
U1 = sqroot( 2 + 2cosx) = sqroot( 2 (1 + cosx))
= ?? (now , how do i simplify)

because , i should have when i have simplified a maximum :

U1 = 2cos(x/2)
= 2 (2cosx² - 1)
= 4cosx² - 2

So how do i go from sqroot( 2 + 2cosx) to 4cosx² - 2 ??

 

[genneth]

Use $$\cos 2x = \cos^2 x - \sin^2 x$$

 

[JPC]

oh ...

i realized i did a mistake here :

this is wrong :
"U1 = 2cos(x/2)
= 2 (2cosx² - 1)
= 4cosx² - 2"

i can't believe , i confused cos(0.5x) and cos(2x)

////

now that i realized i did it like the other way round :

U1 = root( 2 + 2cos(x))
= root(2 + 2(2cos(0.5x)² - 1))
= root(4cos(0.5x)²)
= 2cos(0.5x)

Ok , managed to to do all the rest now , thanks