Proof:


Step 1: Simplify the left-hand side of the equation
- Let's start by simplifying the left-hand side of the equation:
1 + cos∅ + sin∅ / 1 + cos∅ - sin∅
- To simplify this expression, we can multiply both the numerator and the denominator by 1 - cos∅:
(1 + cos∅ + sin∅) * (1 - cos∅) / (1 + cos∅ - sin∅) * (1 - cos∅)
- Expanding both the numerator and the denominator, we get:
(1 - cos^2∅ + sin∅ - cos∅sin∅) / (1 - cos^2∅ + cos∅ - sin∅)
- Simplifying further, we get:
(sin∅ + 1) / (cos∅ - sin∅)

Step 2: Simplify the right-hand side of the equation
- Now, let's simplify the right-hand side of the equation:
1 + sin∅ / cos∅
- To simplify this expression, we can rewrite it as:
(cos∅ + sin∅) / cos∅
- Simplifying further, we get:
(sin∅ + 1) / (cos∅ - sin∅)

Step 3: Compare the simplified expressions
- After simplifying both the left-hand side and the right-hand side of the equation, we see that they are equal:
(sin∅ + 1) / (cos∅ - sin∅) = (sin∅ + 1) / (cos∅ - sin∅)
- Therefore, we have proved that 1 + cos∅ + sin∅ / 1 + cos∅ - sin∅ = 1 + sin∅ / cos∅.