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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
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