https://file.aiquickdraw.com/mathbot/file/1726067319162jnxwb5n9.jpg

The expression in the image reads as follows:

\[
\frac{\sin 50^\circ}{\cos 40^\circ} + \frac{\csc 40^\circ}{\sec 50^\circ} - 4 \cdot \cos 50^\circ \cdot \csc 40^\circ
\]

### Step-by-step simplification:

1. **Use of complementary angles**:
   \[
   \cos(40^\circ) = \sin(50^\circ) \quad \text{since} \quad 50^\circ + 40^\circ = 90^\circ
   \]

2. **First term**:
   \[
   \frac{\sin 50^\circ}{\cos 40^\circ} = \frac{\sin 50^\circ}{\sin 50^\circ} = 1
   \]

3. **Second term**:
   \[
   \frac{\csc 40^\circ}{\sec 50^\circ} = \frac{1/\sin 40^\circ}{1/\cos 50^\circ} = \frac{\cos 50^\circ}{\sin 40^\circ} = 1 \quad \text{(as} \quad \sin 40^\circ = \cos 50^\circ)
   \]

4. **Third term**:
   \[
   -4 \cdot \cos 50^\circ \cdot \csc 40^\circ = -4 \cdot \cos 50^\circ \cdot \frac{1}{\sin 40^\circ} = -4
   \]

### Final Expression:
\[
1 + 1 - 4 = -2
\]

Thus, the simplified value of the given expression is \( \boxed{-2} \).

---

Would you like more details or have any questions?

### Related Questions:
1. How do we derive trigonometric identities using complementary angles?
2. What is the relationship between secant and cosecant?
3. Can we generalize this problem for other angles?
4. How do you use the Pythagorean identity in trigonometric expressions?
5. How do complementary angles simplify trigonometric expressions?

**Tip**: Understanding complementary angles can often simplify trigonometric expressions involving sums of angles that add to \(90^\circ\).

Simplify the expression: \( \frac{\sin 50^\circ}{\cos 40^\circ} + \frac{\csc 40^\circ}{\sec 50^\circ} - 4 \cdot \cos 50^\circ \cdot \csc 40^\circ \)

Learn how to simplify a complex trigonometric expression using complementary angles and trigonometric identities. The expression involves sine, cosine, secant, and cosecant functions, and its final simplified result is -2. This tutorial is suitable for high school students.

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