Given,  tan γ = sec α * sec β + tan α * tan β

=> tan γ = (1/cos α) * (1/cos β) + (sin α/cos α) * (sin β/cos β)

=> tan γ = (1 + sin α * sin β)/(cos α * cos β) 

We know that

       cos 2θ = (1 - tan² θ)/(1 + tan² θ)      

=> cos 2γ = (1 - tan² γ)/(1 + tan² γ)

=> cos 2γ = [1 - {(1 + sin α * sin β)/(cos α * cos β)}² ]/[1 + {(1 + sin α * sin β)/(cos α

* cos cos 2γ =β)}² ]

=> cos 2γ = [(cos α * cos β)² - (1 + sin α * sin β)² ]/[(cos α * cos β)² + (1 + sin α *

sin β)² ]

=> cos 2γ = {cos² α * cos² β - (1 + sin² α * sin² β + 2*sin α * sin β)}/{cos² α * cos² β +

(1 + sin² α * sin² β + 2*sin α * sin β)}

=> cos 2γ = {cos² α * cos² β - 1 - sin² α * sin² β - 2*sin α * sin β)}/{cos² α * cos² β +

1 + sin² α * sin² β + 2*sin α * sin β}

=> cos 2γ = {cos² α * cos² β - (1 - cos² α) * (1 - cos² β) - 2*sin α * sin β - 1}/{cos² α *

cos² β + (1 - cos² α) * (1 - cos² β) + 2*sin α * sin β + 1}

=> cos 2γ = {cos² α * cos² β - (1 - cos² α - cos² β + cos² α * cos² β) - 2*sin α * sin β -

1}/{cos² α * cos² β + (1 - cos² α - cos² β + cos² α * cos² β) + 2*sin α * sin β) + 1}

=> cos 2γ = {cos² α * cos² β - 1 + cos² α + cos² β - cos² α * cos² β) - 2*sin α * sin β -

1}/{cos² α * cos² β + 1 - cos² α - cos² β + cos² α * cos² β + 2*sin α * sin β + 1}

=> cos 2γ = {cos² α + cos² β - 2*sin α * sin β}/{2 - cos² α - cos² β + 2*sin α * sin β}

=> cos 2γ = {1 - sin² α + 1 - sin² β - 2*sin α * sin β}/{1 - cos² α + 1 - cos² β + 2*sin α

* sin β}

=> cos 2γ = {2 - (sin² α + sin² β - 2*sin α * sin β)}/{sin² α + sin² β + 2*sin α* sin β}

=> cos 2γ = {2 - (sin α + sin β)² }/(sin α + sin β)²

=> cos 2γ = 2/(sin α + sin β)² - 1

 Now, cos 2γ is least when sin α and sin β have maximum value.

i.e. sin α = 1 and sin β = 1

=> cos 2γ = 2/(1 + 1)² - 1 

=> cos 2γ = 2/4 - 1

=> cos 2γ = 1/2 - 1

=> cos 2γ = -1/2

Hence, least value of cos 2γ is -1/2