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Let's start by analyzing the given expression:

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

To simplify this expression, we'll use trigonometric identities. First, let's recall the relevant identities:

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. \(\tan(90^\circ - \theta) = \cot(\theta)\)

Substitute these identities into the given expression:

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

Next, we simplify the denominator:

\[ -\tan(\theta) \cdot \cot(\theta) = -1 \]

So, the expression becomes:

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

Simplify by removing the negative sign in the denominator:

\[ \sin(\theta) + \cos(\theta) - \sec(\theta) \]

So, the simplified 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 we simplify trigonometric expressions using identities?
2. What are the basic trigonometric identities and how are they derived?
3. Can you explain how to use the unit circle to understand trigonometric functions?
4. What are the properties of the sine and cosine functions?
5. How do we solve trigonometric equations involving multiple angles?

Tip: Always start by writing down the basic trigonometric identities when simplifying complex expressions. This can help you see which identities might apply and make the process easier.

Learn how to simplify a trigonometric expression involving sine, cosine, and secant using trigonometric identities. This detailed solution covers the use of identities like sin(180° + θ) = -sin(θ) and cos(180° - θ) = -cos(θ). Suitable for advanced high school students studying trigonometry.

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