Question: Prove the identity: cos(3 pie /4+x)-cos((3 pie /4-x)=-root2 sinx.
expand the cosines.
lhs = cos(3 pie /4+x) - cos(3 pie /4-x)
lhs = cos(3π/4).cos(x) - sin(3π/4).sin(x) - {cos(3π/4).cos(x) + sin(3π/4).sin(x) }
lhs = -2.sin(3π/4).sin(x)
and sin(3π/4) = 1/√2, hence
lhs = -√2.sin(x)
But rhs = -√2.sin(x)
i.e. lhs = rhs, for all x.
Therefore the identity is proven