Prove that `cos(x+y)=cosxcosy-sinxsiny`

Prove that cos (x + y) = cos x cos y-sin x sin y

The 2 triangle in the figure above share a common side. What is cos(x+y) ("Note : "cos (x+y)=cos xcosy-sinxsiny" for all x and y.")

Prove geometrically that cos(x + y) = cos x cos y - sinx sin y and hence prove cos((pi)/(2) + x) = - sin x .

Prove that (cos x-cos y)^(2)+(sin x-sin y)^(2)=4 sin ^(2)((x-y)/(2))

Prove that: (cos x+cos y)^(2)+(sin x-sin y)^(2)=4cos^(2)backslash(x+y)/(2)

Prove that (cos x +cos y)^(2)+(sin x-sin y)^(2)=4 cos^(2) ((x+y)/(2))

Prove that: cos(pi/4-x)cos(pi/4-y)-sin(pi/4-x)sin(pi/4-y)=sin(x+y)

Prove that cos(pi/4-x)cos(pi/4-y)-sin(pi/4-x)sin(pi/4-y)=sin(x+y)