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The sine difference formula is a powerful tool in trigonometry that simplifies the difference between the sines of two angles. It is especially useful when dealing with complex angle problems. The formula is expressed as:

• \( \sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \)

This equation helps us transform a subtraction problem into a multiplication, making it easier to solve. By converting the sine difference into a product of sine and cosine, we can leverage known trigonometric values to find our answer. In the given exercise, substituting \(A = 130^{\circ}\) and \(B = 110^{\circ}\) allows us to proceed with simplifying without directly calculating large angle sines. This method often reveals elegant solutions hidden in otherwise complex-looking trigonometric expressions.

To apply the sine difference formula effectively, the angles within the formula need to be simplified. This involves two parts:

• The average of the angles \( \frac{A + B}{2} \)
• The difference divided by two \( \frac{A - B}{2} \)

For example, in our case:

• The average of \(130^{\circ}\) and \(110^{\circ}\) is \(\frac{240^{\circ}}{2} = 120^{\circ}\)
• The difference is \(\frac{20^{\circ}}{2} = 10^{\circ}\)

These simplifications turn an otherwise tedious process of calculating the sine values directly into a manageable task using known trigonometric values. The angles thus simplified assist in accurate calculation while also making demonstration of identities like the one in this exercise clear and straightforward.

Trigonometric values are the backbone of solving trigonometric equations and identities efficiently. These include well-known values for sine and cosine at particular angles, such as 0°, 30°, 45°, 60°, 90°, and so on. In our exercise, utilizing the trigonometric value \( \cos(120^{\circ}) = -\frac{1}{2} \) is crucial. This knowledge allows us to substitute directly into our sine difference formula:- \( 2 \cos(120^{\circ}) \sin(10^{\circ}) = 2 \left(-\frac{1}{2}\right) \sin(10^{\circ}) = -\sin(10^{\circ}) \)This simple substitution turns the problem into a straightforward calculation, enhancing both comprehension and speed. Familiarity with these trigonometric values means students can tackle a broad range of problems they might encounter in their studies, making them less reliant on calculators for solutions and more proficient in recognizing patterns in trigonometric identities.