the question
How to simplify $\sin(x-y)\cos(y)+\cos(x-y)\sin(y)$
my steps
I tried to use trig identities on the $\sin(x-y)$ and $\cos(x-y)$ and tried to distribute the others in but it didn't work. Any ideas?
[Math] How to simplify $\sin(x-y)\cos(y)+\cos(x-y)\sin(y)$
trigonometry
Best Answer
Let's do the (harder) method attempted by the OP, "but it didn't work". $$ \sin(x-y)\cos(y)+\cos(x-y)\sin(y) \\ = \big[\sin(x)\cos(y)-\cos(x)\sin(y)\big]\cos(y)+\big[\cos(x)\cos(y)+\sin(x)\sin(y)\big]\sin(y) \\ = \sin(x)\cos(y)\cos(y)-\cos(x)\sin(y)\cos(y)+\cos(x)\cos(y)\sin(y)+\sin(x)\sin(y)\sin(y) \\= \sin(x)\cos^2(y)+\sin(x)\sin^2(y) \\= \sin(x)\big[\cos^2(y)+\sin^2(y)\big] \\= \sin(x)\big[ 1 \big] \\=\sin(x) $$