Solution:

We are given the equation (1 − tan θ) (1 + sin^2θ) = 1 + tan θ and we need to solve for θ.

First, let's simplify the equation step by step.

Step 1: Expand the brackets on the left side of the equation:

(1 − tan θ) (1 + sin^2θ) = 1 + tan θ
1 + sin^2θ - tan θ - sin^2θ * tan θ = 1 + tan θ

Step 2: Simplify the equation further:

1 - tan θ - sin^2θ * tan θ = tan θ

Step 3: Move all terms to one side of the equation:

1 - tan θ - tan θ - sin^2θ * tan θ = 0
1 - 2tan θ - sin^2θ * tan θ = 0

Step 4: Factor out -tan θ:

1 - 2tan θ - tan θ * (sin^2θ + 1) = 0

Step 5: Recall the identity sin^2θ + cos^2θ = 1:

1 - 2tan θ - tan θ * (cos^2θ) = 0

Step 6: Substitute tan θ = sin θ / cos θ:

1 - 2(sin θ / cos θ) - (sin θ / cos θ) * (cos^2θ) = 0

Step 7: Simplify the equation further:

cos θ - 2sin θ - sin θ * cos θ = 0

Step 8: Move all terms to one side of the equation:

cos θ - 2sin θ - sin θ * cos θ = 0
cos θ - sin θ * (1 + cos θ) = 0

Step 9: Factor out cos θ:

cos θ * (1 - sin θ) - sin θ * (1 + cos θ) = 0

Step 10: Use the zero product property:

cos θ * (1 - sin θ) = 0 or sin θ * (1 + cos θ) = 0

Step 11: Solve for θ in each case:

Case 1: cos θ = 0
This occurs when θ = π/2 or 3π/2.

Case 2: 1 - sin θ = 0
This occurs when sin θ = 1, which gives θ = π/2.

Case 3: 1 + cos θ = 0
This occurs when cos θ = -1, which gives θ = π.

Therefore, the solutions for θ are θ = π/2, 3π/2, and π.