# Difference to Product identities

A trigonometric identity that expresses the transformation of difference of the trigonometric functions into the product form of trigonometric functions is called the difference to product identity.

There are two types of difference to product transformation identities and they are used as formulas in trigonometry. Now, let us learn the difference to product trigonometric identities with proofs.

### Sine functions

$\sin{\alpha}-\sin{\beta}$ $\,=\,$ $2\cos{\Bigg(\dfrac{\alpha+\beta}{2}\Bigg)}\sin{\Bigg(\dfrac{\alpha-\beta}{2}\Bigg)}$

The difference of sine functions can be transformed into the product of the cosine and sine functions. It is called the difference to product transformation identity of the sine functions.

The difference to product identity of sine functions is also written in the following two forms popularly.

$(1). \,\,\,$ $\sin{x}-\sin{y}$ $\,=\,$ $2\cos{\Bigg(\dfrac{x+y}{2}\Bigg)}\sin{\Bigg(\dfrac{x-y}{2}\Bigg)}$

$(2). \,\,\,$ $\sin{C}-\sin{D}$ $\,=\,$ $2\cos{\Bigg(\dfrac{C+D}{2}\Bigg)}\sin{\Bigg(\dfrac{C-D}{2}\Bigg)}$

### Cosine functions

$\cos{\alpha}-\cos{\beta}$ $\,=\,$ $-2\sin{\Bigg(\dfrac{\alpha+\beta}{2}\Bigg)}\sin{\Bigg(\dfrac{\alpha-\beta}{2}\Bigg)}$

The difference of cosine functions can be transformed into the product of the sine functions. It is called the difference to product transformation identity of the cosine functions.

The difference to product identity of cosine functions is also written in the following two forms popularly.

$(1). \,\,\,$ $\cos{x}-\cos{y}$ $\,=\,$ $-2\sin{\Bigg(\dfrac{x+y}{2}\Bigg)}\sin{\Bigg(\dfrac{x-y}{2}\Bigg)}$

$(2). \,\,\,$ $\cos{C}-\cos{D}$ $\,=\,$ $-2\sin{\Bigg(\dfrac{C+D}{2}\Bigg)}\sin{\Bigg(\dfrac{C-D}{2}\Bigg)}$

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The text passage provides trigonometric identities for transforming the difference of sine and cosine functions into their product forms. It also includes alternative forms of these identities.
no_question

the passage does not contain any mathematical questions or answers to such questions. it is purely expository, providing definitions and identities.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Difference to Product identities

A trigonometric identity that expresses the transformation of difference of the trigonometric functions into the product form of trigonometric functions is called the difference to product identity.

There are two types of difference to product transformation identities and they are used as formulas in trigonometry. Now, let us learn the difference to product trigonometric identities with proofs.

### Sine functions

$\sin{\alpha}-\sin{\beta}$ $\,=\,$ $2\cos{\Bigg(\dfrac{\alpha+\beta}{2}\Bigg)}\sin{\Bigg(\dfrac{\alpha-\beta}{2}\Bigg)}$

The difference of sine functions can be transformed into the product of the cosine and sine functions. It is called the difference to product transformation identity of the sine functions.

The difference to product identity of sine functions is also written in the following two forms popularly.

$(1). \,\,\,$ $\sin{x}-\sin{y}$ $\,=\,$ $2\cos{\Bigg(\dfrac{x+y}{2}\Bigg)}\sin{\Bigg(\dfrac{x-y}{2}\Bigg)}$

$(2). \,\,\,$ $\sin{C}-\sin{D}$ $\,=\,$ $2\cos{\Bigg(\dfrac{C+D}{2}\Bigg)}\sin{\Bigg(\dfrac{C-D}{2}\Bigg)}$

### Cosine functions

$\cos{\alpha}-\cos{\beta}$ $\,=\,$ $-2\sin{\Bigg(\dfrac{\alpha+\beta}{2}\Bigg)}\sin{\Bigg(\dfrac{\alpha-\beta}{2}\Bigg)}$

The difference of cosine functions can be transformed into the product of the sine functions. It is called the difference to product transformation identity of the cosine functions.

The difference to product identity of cosine functions is also written in the following two forms popularly.

$(1). \,\,\,$ $\cos{x}-\cos{y}$ $\,=\,$ $-2\sin{\Bigg(\dfrac{x+y}{2}\Bigg)}\sin{\Bigg(\dfrac{x-y}{2}\Bigg)}$

$(2). \,\,\,$ $\cos{C}-\cos{D}$ $\,=\,$ $-2\sin{\Bigg(\dfrac{C+D}{2}\Bigg)}\sin{\Bigg(\dfrac{C-D}{2}\Bigg)}$

###### Math Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Practice now

###### Math Videos

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The text passage provides trigonometric identities for transforming the difference of sine and cosine functions into their product forms. It also includes alternative forms of these identities.

TYPE: NO_QUESTION

The passage does not contain any mathematical questions or answers to such questions. It is purely expository, providing definitions and identities.