Proving the value of an indefinite integral

Question

If $$ I = \int{\frac{(\sin(x)-\cos(x))dx}{(\sin(x) + \cos(x))(\sqrt{\sin(x)\cos(x) + \sin^2(x)\cos^2(x)}) )}} = \csc^{-1}g(x) +c $$ then prove that $$ g(x) = 1 + \sin(2x)$$

My approach : I assumed $ \sin(x) + \cos(x) = z$ so that the numerator can be substituted as $ (\sin(x) - \cos(x))dx = -dz$. But how do I reach the proof?

Any help will be appreciated.

Answer 1

Hint multiply and divide by $\sin (x)+\cos (x) $ . Now use the fact that $\sin^2 (x)-\cos^2 (x)=-cos (2x), (\sin (x)+\cos(x))^2=1+\sin (2x) ,2\sin (x)\cos (x)=\sin (2x) $ . Also you can manipulate the root part as $\frac {2\sin (2x)}{4}+\frac {\sin^2 (2x)}{4} $ I hope you can continue from here.

Answer 2

HINT:

As $\int(\sin x-\cos x)dx=-(\cos x+\sin x)$

set $\cos x+\sin x=u\implies2\sin x\cos x=u^2-1$