https://file.aiquickdraw.com/mathbot/file/1729336468228b06c5tr8.jpg

The equation in the image reads:

\[
\frac{\cot \theta}{\cot \theta - \cot 30^\circ} + \frac{\tan \theta}{\tan \theta - \tan 30^\circ} \quad \text{is equal to:}
\]

Let's break this down step by step and simplify it:

1. **Evaluate known values:**
   - \(\cot 30^\circ = \sqrt{3}\)
   - \(\tan 30^\circ = \frac{1}{\sqrt{3}}\)

So the equation becomes:

\[
\frac{\cot \theta}{\cot \theta - \sqrt{3}} + \frac{\tan \theta}{\tan \theta - \frac{1}{\sqrt{3}}}
\]

2. **Simplify the terms:**

Both terms in the expression are of a similar form, but with reciprocal trigonometric functions (cotangent and tangent). We can try combining them after simplifying further.

Next, simplify the overall expression by finding common denominators for each term if needed, or using trigonometric identities to relate \(\cot \theta\) and \(\tan \theta\) (since \(\cot \theta = \frac{1}{\tan \theta}\)).

Would you like to proceed with a detailed step-by-step breakdown of the solution?

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Here are five questions to extend the problem:
1. What is the relationship between \(\cot \theta\) and \(\tan \theta\)?
2. How do you convert degrees to radians for trigonometric functions?
3. Can we use trigonometric identities to simplify the expressions further?
4. How would the equation change if we used other angles like 45° or 60° instead of 30°?
5. What are the key properties of cotangent and tangent at common angles?

**Tip:** Using known values for trigonometric functions of special angles can help simplify expressions quickly.

Simplify the expression \(\frac{\cot \theta}{\cot \theta - \cot 30^\circ} + \frac{\tan \theta}{\tan \theta - \tan 30^\circ}\).

Learn how to simplify a trigonometric expression involving cotangent and tangent functions at special angles. This problem is suitable for advanced trigonometry students in Grades 11-12 and covers key identities and special angle values.

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