https://r2.erweima.ai/mathbot/image/1726583792430fdk3pnw7.jpg

The problem in the image asks to calculate the value of \( x \) in the following trigonometric equation:

\[
2x \cdot \sin 30^\circ - \tan 45^\circ = \sec 60^\circ - x
\]

### Step-by-step solution:

1. **Evaluate the trigonometric values**:
   - \( \sin 30^\circ = \frac{1}{2} \)
   - \( \tan 45^\circ = 1 \)
   - \( \sec 60^\circ = \frac{1}{\cos 60^\circ} = \frac{1}{\frac{1}{2}} = 2 \)

2. **Substitute these values into the equation**:

\[
2x \cdot \frac{1}{2} - 1 = 2 - x
\]

This simplifies to:

\[
x - 1 = 2 - x
\]

3. **Solve for \( x \)**:
   - Add \( x \) to both sides:
   \[
   x + x - 1 = 2
   \]
   \[
   2x - 1 = 2
   \]
   - Add 1 to both sides:
   \[
   2x = 3
   \]
   - Divide by 2:
   \[
   x = \frac{3}{2} = 1.5
   \]

Thus, the value of \( x \) is \( 1.5 \).

---

Do you want any further details or have any questions about the process?

Here are some related questions to consider:
1. How do you calculate the values of other common trigonometric angles?
2. What are the definitions of sine, tangent, and secant functions?
3. How do you solve trigonometric equations with multiple unknowns?
4. Can this method be used to solve more complex trigonometric identities?
5. What is the unit circle and how does it relate to trigonometric functions?

**Tip**: Always double-check trigonometric values for standard angles like \( 30^\circ, 45^\circ, \) and \( 60^\circ \) to avoid simple calculation errors.

2x · sin(30°) - tan(45°) = sec(60°) - x

This problem involves solving a trigonometric equation with the functions sin, tan, and sec. The key steps include evaluating standard trigonometric values for 30°, 45°, and 60° and applying algebraic techniques to isolate x. Suitable for Grades 9-10 students learning about trigonometry.

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