Question

Question #a927e

Answer 1

We know

`cos2theta=cos^2theta-sin^2theta`

`cos2theta=cos^2theta-(1-cos^2theta)`

`cos2theta=2cos^2theta-1`

`cos^2theta=1/2*(1+cos2theta)`

`costheta=sqrt(1/2(1+cos2*theta))`

putting `theta=22.5` we get

`cos(22.5)=sqrt(1/2(1+cos45^o)`

`cos(22.5)=sqrt(1/2(1+1/sqrt2)`

Answer 2

`sqrt(2 + sqrt2)/2`

Explanation:

Call cos (22.5) = cos t
`cos 2t = cos (45) = sqrt2/2`
Use trig identity:
`2cos^2 t = 1 + cos 2t = 1 + sqrt2/2 = (2 + sqrt2)/2`
`cos^2 t = (2 + sqrt2)/4`
`cos t = +- sqrt(2 + sqrt2)/2`
Since cos 22.5 is positive, then, take the positive answer.
`cos (22.5) = cos t = sqrt(2 +sqrt2)/2`