If sec(φ+α)+sec(φ-α)=2secφ show that cosφ=+-√cosα/2

Steps to prove cosφ=+-√cosα/2

From Question

⇒sec(φ+α)+sec(φ-α)=2secφ

⇒1/cos(φ+α)+1/cos(φ-α)=2/cosφ

⇒{cos(φ-α)+cos(φ+α)}/{cos(φ-α).cos(φ+α)}=2/cosφ

⇒2.cosφ.cosα/(cos^2φ-sin^2α)=2/cosφ

⇒cos^2φ.cosα=cos^2φ-sin^2α

⇒cos^2φ.cosα-cos^2φ=-sin^2α

⇒cos^2φ.cosα-cos^2φ=-(1-cos^2α)

⇒cos^2φ(cosα-1)=(cosα+1)(cosα-1)

⇒cos^2φ=1+cosα

⇒cos^2φ=2cos^2α/2

⇒cosφ=+-√cosα/2

Hence cosφ=+-√cosα/2 is proved

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