What is the right way of simplifying this trigonometric expression and why?

Question

Problem: $$\frac{\frac{cosA}{sinA}-cosA}{\frac{cosA}{sinA}+cosA}$$

sol1:$$=\frac{cosA\left(\frac{1}{sinA}-1\right)}{cosA\left(\frac{1}{sinA}+1\right)}$$ $$=\frac{\left(\frac{1}{sinA}-1\right)}{\left(\frac{1}{sinA}+1\right)}$$ $$=\frac{cosecA-1}{cosecA+1}$$

sol2: $$=\frac{cosA-sinAcosA}{cosA+sinAcosA}$$ $$By ÷cosA$$ $$=\frac{1-cosecA}{1+cosecA}$$

Which is the right answer and why?

Answer 1

Your last step is incorrect.

$$\dfrac{1-\sin A}{1+\sin A}=\dfrac{\csc A-1}{\csc A+1}$$

Answer 2

your sol2 should be:

$$=\frac{cosA−sinAcosA}{cosA+sinAcosA}$$

dividing by cosA gives,

$$=\frac{\frac{cosA}{cosA}-\frac{sinACosA}{cosA}}{\frac{cosA}{cosA}+\frac{sinAcosA}{cosA}}$$

the result is ,

$$=\frac{1-sinA}{1+sinA}$$

focus on your mistake.