If sec(theta+alpha) + sec(theta-alpha)=2sectheta prove that cos theta= 2^1/2 cosalpha/2

sec(θ+α)+sec(θ−α)=2sec(θ) can be written as

1cos(θ+α)+1cos(θ−α)=2cosθ

or cos(θ+α)+cos(θ−α)cos(θ+α)cos(θ−α)=2cos(θ)

or2×2cosθcosαcos(θ+α+θ−α)+cos(θ+α−θ+α)=2cosθ

or 2cosθcosαcos2θ+cos2α=1cosθ

or 2cos2θcosα=cos2θ+cos2α

or 2cos2θcosα=2cos2θ−1+2cos2α−1

or 2cos2θ(cosα−1)=2cos2α−2

or cos2θ=cos2α−1cosα−1=cosα+1

or cos2θ=2cos2(α2)

and cosθ=√2cos(α2)

Hope it helps