https://file.aiquickdraw.com/mathbot/file/1725895825311hrjenpz5.jpg

Let's solve the expression provided in the image:

### Expression:
\[
\frac{\frac{4}{\sqrt{3}} \sin 60^\circ + \frac{\sqrt{2}}{2} \sec 45^\circ - \sqrt{3} \tan 30^\circ}{\frac{2}{\sqrt{3}} \cot 60^\circ - \frac{2}{\sqrt{3}} \csc 60^\circ + 5 \cos 0^\circ}
\]

### Step 1: Evaluate Trigonometric Functions

1. **\(\sin 60^\circ\)**:
   \[
   \sin 60^\circ = \frac{\sqrt{3}}{2}
   \]
   
2. **\(\sec 45^\circ\)**:
   \[
   \sec 45^\circ = \frac{1}{\cos 45^\circ} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}
   \]

3. **\(\tan 30^\circ\)**:
   \[
   \tan 30^\circ = \frac{1}{\sqrt{3}}
   \]

4. **\(\cot 60^\circ\)**:
   \[
   \cot 60^\circ = \frac{1}{\tan 60^\circ} = \frac{1}{\sqrt{3}}
   \]

5. **\(\csc 60^\circ\)**:
   \[
   \csc 60^\circ = \frac{1}{\sin 60^\circ} = \frac{2}{\sqrt{3}}
   \]

6. **\(\cos 0^\circ\)**:
   \[
   \cos 0^\circ = 1
   \]

### Step 2: Substitute the values into the expression

\[
\text{Numerator: } \frac{4}{\sqrt{3}} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \sqrt{2} - \sqrt{3} \cdot \frac{1}{\sqrt{3}}
\]

\[
= \frac{4 \cdot \sqrt{3}}{2 \cdot \sqrt{3}} + \frac{2}{2} - 1
\]

\[
= 2 + 1 - 1 = 2
\]

\[
\text{Denominator: } \frac{2}{\sqrt{3}} \cdot \frac{1}{\sqrt{3}} - \frac{2}{\sqrt{3}} \cdot \frac{2}{\sqrt{3}} + 5 \cdot 1
\]

\[
= \frac{2}{3} - \frac{4}{3} + 5
\]

\[
= \frac{-2}{3} + 5 = \frac{-2 + 15}{3} = \frac{13}{3}
\]

### Step 3: Calculate the final value

\[
\frac{\text{Numerator}}{\text{Denominator}} = \frac{2}{\frac{13}{3}} = \frac{2 \times 3}{13} = \frac{6}{13}
\]

### Final Answer:
\[
\boxed{\frac{6}{13}}
\]

Would you like more details on any of these steps? Here are some questions for further practice:

1. How do you evaluate other trigonometric functions for special angles?
2. What are the Pythagorean identities and how are they used in trigonometry?
3. Can you simplify expressions involving trigonometric identities?
4. What is the significance of the unit circle in trigonometry?
5. How do you convert between degrees and radians?

**Tip**: Always double-check the values of trigonometric functions for standard angles to avoid errors in calculations.

Evaluate the following expression: 
\(\frac{\frac{4}{\sqrt{3}} \sin 60^\circ + \frac{\sqrt{2}}{2} \sec 45^\circ - \sqrt{3} \tan 30^\circ}{\frac{2}{\sqrt{3}} \cot 60^\circ - \frac{2}{\sqrt{3}} \csc 60^\circ + 5 \cos 0^\circ}\)

Learn how to evaluate a trigonometric expression involving sine, secant, tangent, cotangent, cosecant, and cosine of special angles. This problem simplifies trigonometric identities and involves basic simplification techniques. Suitable for Grades 10-12.

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