How do you express $cos(4theta)$ in terms of $cos(2theta)$ ?

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Answer 1 · 2015-06-10T23:05:38

$cos(4theta) = 2(cos(2theta))^2-1$

Start by replacing $4theta$ with $2theta+2theta$

$cos(4theta) = cos(2theta+2theta)$

Knowing that $cos(a+b) = cos(a)cos(b)-sin(a)sin(b)$ then

$cos(2theta+2theta) = (cos(2theta))^2-(sin(2theta))^2$

Knowing that $(cos(x))^2+(sin(x))^2 = 1$ then

$(sin(x))^2 = 1-(cos(x))^2$

$rarr cos(4theta) = (cos(2theta))^2-(1-(cos(2theta))^2)$

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

How do you express cos(4theta)cos(4theta) in terms of cos(2theta)cos(2theta)?