$x=2\sin A/(1+\cos A+\sin A)$ , then $ (1-\cos A- \sin A)/\cos A=$?

Question

$x=\cfrac{2\sin A}{1+\cos A+\sin A}$ , then $ \cfrac{1-\cos A- \sin A}{\cos A}=?$

Options are

$1$. $x$

$2$. $\cfrac 1x$

$3$. $-x$

$4$. $\cfrac{-1}x$

Can not attend it.

I don't know how to approach it.

Answer 1

Let $$ y ={\cos A \over 1-\cos A- \sin A}$$ We are interested in ${1\over y}$. Then $$xy = {2\sin A\cos A \over 1-(\cos A+\sin A)^2 } ={\sin 2A \over 1-\cos ^2A -\sin^2 A -2\sin A \cos A } =- 1$$

So ${1\over y} = -x$

Answer 2

$x=\dfrac{2sinA}{1+cosA+sinA}.\left(\dfrac{1-cosA-sinA}{1-cosA-sinA}\right)$

$x=\dfrac{{2sinA}.({1-cosA-sinA})}{1-(cosA+SinA)^{2}}=-\dfrac{({1-cosA-sinA})}{cosA}$

$\implies\dfrac{({1-cosA-sinA})}{cosA}=-x$

Answer 3

$$(\cos A+\sin A+1)(\cos A+\sin A-1)=\cdots=2\sin A\cos A$$