Question

Simplify cos(x - y) - cos(x + y)

Answer 1

Hello there. To solve this question, we'll have to remember some properties about the sum-to-product formula.

Given the trigonometric expression:

`\cos(x-y)-\cos(x+y)`

We have to simplify it.

First, remember the angle sum formulas:

`\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}`

Subtracting the second equation from the second equation, we'll have:

`\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}`

Add the terms

`\cos(x-y)-\cos(x+y)=2\sin(x)\sin(y)`

This is also known as one of the sum-to-product formulas;