Prove that :1-cos theta +sin theta / cos theta + sin theta -1 = 1+ sin theta / cos theta

Question

A trigonometry question. I don’t know if there is anything wrong in the question.So please do see to it.

Answer 1

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.