Let's examine the given expression again and simplify it step by step:

$\frac{\sin(180^\circ + \theta) + \cos(180^\circ - \theta) + \csc(90^\circ - \theta)}{\tan(180^\circ - \theta) \cdot \tan(90^\circ - \theta)}$

Using the trigonometric identities, we have:

1. $\sin(180^\circ + \theta) = -\sin(\theta)$
2. $\cos(180^\circ - \theta) = -\cos(\theta)$
3. $\csc(90^\circ - \theta) = \sec(\theta)$
4. $\tan(180^\circ - \theta) = -\tan(\theta)$
5. $\tan(90^\circ - \theta) = \cot(\theta)$

Now substitute these identities into the expression:

$\frac{-\sin(\theta) - \cos(\theta) + \sec(\theta)}{-\tan(\theta) \cdot \cot(\theta)}$

Simplify the denominator:

$-\tan(\theta) \cdot \cot(\theta) = -\tan(\theta) \cdot \frac{1}{\tan(\theta)} = -1$

So, the expression becomes:

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

Removing the negative sign in the denominator:

$\sin(\theta) + \cos(\theta) - \sec(\theta)$

Thus, the simplified form of the given expression is:

$\sin(\theta) + \cos(\theta) - \sec(\theta)$

Would you like more details or have any questions about this simplification?

Here are some additional questions you might be interested in:

1. How do you derive the trigonometric identities for sum and difference of angles?
2. What is the significance of the Pythagorean identities in trigonometry?
3. Can you explain the relationship between trigonometric functions and their reciprocals?
4. How can you use trigonometric identities to solve real-world problems?
5. What are the properties of the tangent and cotangent functions?

Tip: When simplifying trigonometric expressions, it's often helpful to express all terms in terms of sine and cosine. This can make it easier to see how to combine terms and apply identities.