Question

Verify the identity: Sin(u+y+c)=sinucosycosc+cosusinycosc+cosucosysinc-sinusinysinc

Answer 1

use
sin(A+B)=sinA cosB+sinB cosA

Answer 2

in your problem let
A=u
B=y+c

Answer 3

\(\sin(a+b) = \sin(a)\cos(b)+\sin(b)\cos(a)\)
\(\cos(a+b) = \cos(a)\cos(b)+\sin(a)\sin(b)\)

so you have

\(\sin(u+y+c) = \sin((u+y)+c) = \sin(u+y)\cos(c) + \sin(c)\cos(u+y) =\\ \cos(c)(\sin(u)\cos(y)+\sin(y)\cos(u))+\sin(c)(\cos(u)\cos(y)+\sin(u)\sin(y)) =\\ \small \cos(c)\sin(u)\cos(y)+\cos(c)\sin(y)\cos(u)+\sin(c)\cos(u)\cos(y)+\sin(c)\sin(u)\sin(y)\)

as desired

Answer 4

that was not fun

Answer 5

Thank you so much. To both of you.