Cos alpha minus beta =1 and cos alpha plus beta =1 /e then ordered pair of alpha and beta is

We are given the equations:

cos(α - β) = 1
cos(α + β) = 1/e
Step 1: Solve for α - β
From equation (1),
cos(α - β) = 1
This implies:
α - β = 2nπ, where n is any integer.

Step 2: Solve for α + β
From equation (2),
cos(α + β) = 1/e

Taking the inverse cosine on both sides:
α + β = cos⁻¹(1/e)

Step 3: Solve for α and β
We now have the system of equations:

α - β = 2nπ
α + β = cos⁻¹(1/e)
Adding both equations:
(α - β) + (α + β) = 2nπ + cos⁻¹(1/e)
2α = 2nπ + cos⁻¹(1/e)
α = nπ + (1/2) cos⁻¹(1/e)

Subtracting the second equation from the first:
(α - β) - (α + β) = 2nπ - cos⁻¹(1/e)
-2β = 2nπ - cos⁻¹(1/e)
β = -nπ + (1/2) cos⁻¹(1/e)

Final Answer
The ordered pair (α, β) is:

( nπ + (1/2) cos⁻¹(1/e), -nπ + (1/2) cos⁻¹(1/e) ) where n is any integer.