Respuesta :
Hello there. To solve this question, we'll have to remember some properties about the sum-to-product formula.
Given the trigonometric expression:
[tex]\cos(x-y)-\cos(x+y)[/tex]We have to simplify it.
First, remember the angle sum formulas:
[tex]\begin{gathered} \cos(x-y)=\cos(x)\cos(y)+\sin(x)\sin(y) \\ \cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y) \end{gathered}[/tex]Subtracting the second equation from the second equation, we'll have:
[tex]\begin{gathered} \cos(x-y)-\cos(x+y)=\cos(x)\cos(y)+\sin(x)\sin(y)-(\cos(x)\cos(y)-\sin(x)\sin(y)) \\ \\ \cos(x-y)-\cos(x-y)=\cos(x)\cos(y)+\sin(x)\sin(y)-\cos(x)\cos(y)+\sin(x)\sin(y) \end{gathered}[/tex]Add the terms
[tex]\cos(x-y)-\cos(x+y)=2\sin(x)\sin(y)[/tex]This is also known as one of the sum-to-product formulas;
