Answer:
See Below.
Step-by-step explanation:
We want to verify:
[tex]\cos(x-y)-\cos(x+y)=2\sin(x)\sin(y)[/tex]
We will utilize the following identities:
[tex]\cos(x-y)=\cos(x)\cos(y)+\sin(x)\sin(y)[/tex]
And:
[tex]\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)[/tex]
So, by substitution, we acquire:
[tex](\cos(x)\cos(y)+\sin(x)\sin(y))-(\cos(x)\cos(y)-\sin(x)\sin(y))=2\sin(x)\sin(y)[/tex]
Distribute:
[tex]\cos(x)\cos(y)+\sin(x)\sin(y)-\cos(x)\cos(y)+\sin(x)\sin(y)=2\sin(x)\sin(y)[/tex]
The first and third term will cancel:
[tex]\sin(x)\sin(y)+\sin(x)\sin(y)=2\sin(x)\sin(y)[/tex]
Combine like terms:
[tex]2\sin(x)\sin(y)\stackrel{\checkmark}{=}2\sin(x)\sin(y)[/tex]
