Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It has various formulas that are essential for solving problems related to triangles and periodic phenomena. One such formula that is frequently used in trigonometry is the product-to-sum formula: **2 sin A sin B**. This formula plays a significant role in simplifying trigonometric expressions and solving trigonometric equations. In this article, we will delve into the depths of this formula, understand its derivation, applications, and solve some examples to illustrate its usage effectively.

### Understanding the Product-to-Sum Formula: 2 Sin A Sin B

**1. Derivation of the Formula:**

The product-to-sum formula for **2 sin A sin B** is derived from the trigonometric identity of the cosine of the sum of two angles. By using the trigonometric identity **cos (A + B) = cos A cos B – sin A sin B**, and rearranging the terms, we can get the formula **2 sin A sin B** as follows:

[2 \cdot \left( \frac{1}{2} \cdot (\cos (A – B) – \cos (A + B)) \right)]

[= 2 \cdot \left( \frac{1}{2} \cdot \left( \cos A \cdot \cos B + \sin A \cdot \sin B – (\cos A \cdot \cos B – \sin A \cdot \sin B) \right) \right)]

[= 2 \cdot \left( \frac{1}{2} \cdot \left( \cos A \cdot \cos B + \sin A \cdot \sin B – \cos A \cdot \cos B + \sin A \cdot \sin B \right) \right)]

[= \sin A \cdot \sin B]

Therefore, the formula **2 sin A sin B** is obtained.

**2. Application of the Formula:**

The **2 sin A sin B** formula is particularly useful in trigonometry while simplifying trigonometric expressions involving the product of sines of angles. It allows us to transform products of trigonometric functions into sums or differences, making it easier to manipulate and solve trigonometric equations.

**3. Examples of Using the Formula:**

Let’s consider an example to demonstrate the application of the **2 sin A sin B** formula:

**Example:** Simplify the expression **4 sin 3x sin 2x**.

[4 sin 3x sin 2x = 2(2 sin 3x sin 2x)]

Using the formula **2 sin A sin B = cos (A – B) – cos (A + B)**, we get:

[2(2 sin 3x sin 2x) = 2(cos (3x – 2x) – cos (3x + 2x))]

[= 2(cos x – cos 5x)]

[= 2 cos x – 2 cos 5x]

### Frequently Asked Questions (FAQs):

**Q1: What is the significance of the product-to-sum formula 2 sin A sin B in trigonometry?**

A1: The formula **2 sin A sin B** is significant as it helps in simplifying trigonometric expressions involving the product of sines of angles, making it easier to manipulate and solve trigonometric equations.

**Q2: Can the product-to-sum formula be applied to other trigonometric functions like cosine or tangent?**

A2: Yes, similar product-to-sum formulas exist for other trigonometric functions like cosine and tangent, allowing for the transformation of products into sums or differences.

**Q3: How can the formula 2 sin A sin B be derived using other trigonometric identities?**

A3: The formula **2 sin A sin B** can be derived from the cosine of the sum of two angles trigonometric identity by rearranging the terms to isolate the product of sines.

**Q4: In what types of trigonometric problems is the 2 sin A sin B formula commonly used?**

A4: The formula **2 sin A sin B** is commonly used in trigonometric problems involving simplification of trigonometric expressions, verification of trigonometric identities, and solving trigonometric equations.

**Q5: Are there alternative ways to simplify products of sines of angles if the 2 sin A sin B formula is not applicable?**

A5: Yes, there are other trigonometric identities and formulas that can be used to simplify products of sines of angles, such as the double angle formulas or the sum-to-product formulas.

In conclusion, the product-to-sum formula **2 sin A sin B** is a valuable tool in trigonometry for simplifying trigonometric expressions and solving trigonometric equations. By understanding its derivation, applications, and practicing with examples, one can enhance their proficiency in trigonometric calculations and problem-solving.