text: # 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
  
  The math videos tutorials with visual graphics to learn every concept.
  
  Watch now
  
  ###### Subscribe us
  
  Get the latest math updates from the Math Doubts by subscribing us.
explanation: The provided text is an explanation of trigonometric identities that transform the difference of trigonometric functions into their product forms. It includes the formulas for the difference to product identities for sine and cosine functions, along with their alternative forms. The text does not contain any specific math questions or answers to questions.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 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
  
  The math videos tutorials with visual graphics to learn every concept.
  
  Watch now
  
  ###### Subscribe us
  
  Get the latest math updates from the Math Doubts by subscribing us.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is an explanation of trigonometric identities that transform the difference of trigonometric functions into their product forms. It includes the formulas for the difference to product identities for sine and cosine functions, along with their alternative forms. The text does not contain any specific math questions or answers to questions.
  
  Label: no-question