I spend some time trying to figure out how to prove the following identity:
$$ \cos(x+y)+\sin(x-y)=2 \sin\left(x+\frac{\pi}{4}\right) \sin\left(\frac{\pi}{4}-y\right) $$
I tried to use the following identities:
$$ \cos(x+y)=\cos(x) \sin(y) - \sin(x) \sin(y) $$
and
$$ \sin(x-y)=\sin(x) \cos(y) - \cos(x) \sin(y). $$
After that, I wanted to use
$$ \cos(x) = \sin\left(x+ \frac{\pi}{2}\right). $$
Unfortunately I can't reach the correct identity. Is there another way of doing it ? Thank you.