When θ=π/6 the left-hand side is (1-√3/2+1/2)/(√3/2+1/2-1)=(3/2-√3/2)/(√3/2-1/2)=√3.
The right-hand side is (3/2)/(√3/2)=√3. So the equation could be an identity.
Now to prove if it is:
cosθ(1-cosθ+sinθ)=(1+sinθ)(cosθ+sinθ-1),
cosθ-cos²θ+sinθcosθ=
cosθ+sinθ-1+sinθcosθ+sin²θ-sinθ=cosθ-cos²θ+sinθcosθ, because -1+sin²θ=-cos²θ and sinθ cancels out. Therefore the two sides of the equation are identical.