To simplify the given expression, we start with the original expression:

$\frac{1}{\cot \theta (\sec \theta - 1)} + \frac{\tan \theta}{\sec \theta + 1}$

Step 1: Express in terms of sine and cosine

Recall that: $\cot \theta = \frac{\cos \theta}{\sin \theta}, \quad \tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \text{and} \quad \sec \theta = \frac{1}{\cos \theta}$

Substitute these into the expression:

$\frac{1}{\frac{\cos \theta}{\sin \theta} \left(\frac{1}{\cos \theta} - 1\right)} + \frac{\frac{\sin \theta}{\cos \theta}}{\frac{1}{\cos \theta} + 1}$

Simplify the terms inside the fractions:

1. For the first term:

$\frac{1}{\frac{\cos \theta}{\sin \theta} \left(\frac{1 - \cos \theta}{\cos \theta}\right)} = \frac{1}{\frac{\cos \theta (1 - \cos \theta)}{\sin \theta \cos \theta}} = \frac{\sin \theta}{1 - \cos \theta}$

2. For the second term:

$\frac{\frac{\sin \theta}{\cos \theta}}{\frac{1 + \cos \theta}{\cos \theta}} = \frac{\sin \theta}{1 + \cos \theta}$

Now the expression becomes:

$\frac{\sin \theta}{1 - \cos \theta} + \frac{\sin \theta}{1 + \cos \theta}$

Step 2: Combine the fractions

To add the two fractions, we need a common denominator:

$\frac{\sin \theta (1 + \cos \theta) + \sin \theta (1 - \cos \theta)}{(1 - \cos \theta)(1 + \cos \theta)}$

Simplify the numerator:

$\sin \theta (1 + \cos \theta) + \sin \theta (1 - \cos \theta) = \sin \theta (1 + \cos \theta + 1 - \cos \theta) = \sin \theta (2)$

So, the expression simplifies to:

$\frac{2\sin \theta}{1 - \cos^2 \theta}$

Since $1 - \cos^2 \theta = \sin^2 \theta$ (using the Pythagorean identity), we get:

$\frac{2\sin \theta}{\sin^2 \theta} = \frac{2}{\sin \theta}$

Final Answer:

$\frac{2}{\sin \theta} = 2 \csc \theta$

So, the given expression simplifies to $\boxed{2 \csc \theta}$.

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you simplify trigonometric identities using Pythagorean identities?
2. What are the reciprocal trigonometric functions, and how are they used in simplifications?
3. Can you explain the process of finding a common denominator in rational expressions?
4. What are some key trigonometric identities that are frequently used in simplifications?
5. How can we express other trigonometric functions in terms of sine and cosine?

Tip: Always check for Pythagorean identities when simplifying trigonometric expressions, as they often provide a simpler form.