Question

prove sin (x-y) cos x-cos (x-y) sin x = -sin y

Answer 1

$\underline{\textbf{To prove:}}$

$\mathsf{sin(x-y)\;cos\,x-cos(x-y)sin\,x=-sin\,y}$

$\underline{\textbf{Solution:}}$

$\underline{\textbf{Identity used:}}$

$\boxed{\mathsf{sin(A-B)=\;sin\,A\;cos\,B-cos\,A\;sin\,B}}$

$\mathsf{Consider,}$

$\mathsf{sin(x-y)\;cos\,x-cos(x-y)sin\,x}$

${\textsf{Using the above identity, we get}}$

$\mathsf{=sin(x-y-x)}$

$\mathsf{=sin(-y)}\;\;\;\mathsf{(\because\;sin(-\theta)=-sin\theta)}$

$\mathsf{=-sin\,y}$

$\implies\boxed{\mathsf{sin(x-y)\;cos\,x-cos(x-y)sin\,x=-sin\,y}}$