**Proof: cot 2θ - cosec 2θ = (1 - cos θ) / (1 + cos θ)**

To prove the given equation, we will start by expressing cot 2θ and cosec 2θ in terms of sin and cos functions. Then we will simplify the equation step by step to reach the desired result.

**Step 1: Express cot 2θ and cosec 2θ in terms of sin and cos**

Recall the following trigonometric identities:
- cot θ = cos θ / sin θ
- cosec θ = 1 / sin θ

Using these identities, we can express cot 2θ and cosec 2θ as follows:
- cot 2θ = cos 2θ / sin 2θ
- cosec 2θ = 1 / sin 2θ

**Step 2: Simplify cot 2θ and cosec 2θ**

To simplify cot 2θ, we need to express cos 2θ and sin 2θ in terms of cos θ and sin θ. We can use the double-angle identities for cosine and sine:

- cos 2θ = cos² θ - sin² θ
- sin 2θ = 2sin θ cos θ

Substituting these values, we get:
- cot 2θ = (cos² θ - sin² θ) / (2sin θ cos θ)
= (cos² θ - sin² θ) / (2sin θ cos θ) * (1/cos θ) / (1/cos θ)
= (cos θ - sin θ)(cos θ + sin θ) / (2sin θ cos θ) * (1/cos θ)
= (cos θ - sin θ) / 2sin θ

To simplify cosec 2θ, we can use the reciprocal identity:
- cosec θ = 1 / sin θ

Applying this to cosec 2θ, we have:
- cosec 2θ = 1 / (2sin θ cos θ)
= 1 / 2sin θ * 1 / cos θ
= 1 / 2sin θ * cos θ / cos θ
= cos θ / 2sin θ

**Step 3: Substitute cot 2θ and cosec 2θ back into the equation**

Now that we have simplified cot 2θ and cosec 2θ, we substitute them back into the original equation:
cot 2θ - cosec 2θ = (1 - cos θ) / (1 + cos θ)

Substituting the simplified forms, we have:
((cos θ - sin θ) / 2sin θ) - (cos θ / 2sin θ) = (1 - cos θ) / (1 + cos θ)

**Step 4: Simplify the equation**

To simplify the equation further, we need to find a common denominator for the left side:
((cos θ - sin θ) - cos θ) / 2sin θ = (1 - cos θ) / (1 +