The value of

(c) cosec x

We have,

$\frac{2\left(\sin 2 x+2 \cos ^{2} x-1\right)}{\cos x-\sin x-\cos 3 x+\sin 3 x}$

$=\frac{2(\sin 2 x+\cos 2 x)}{\cos x-\sin x-4 \cos ^{3} x+3 \cos x+3 \sin x-4 \sin ^{3} x}$

$=\frac{2(\sin 2 x+\cos 2 x)}{4 \cos x-4 \cos ^{3} x+2 \sin x-4 \sin ^{3} x}$

$=\frac{2(\sin 2 x+\cos 2 x)}{4 \cos x\left(1-\cos ^{2} x\right)+2 \sin x\left(1-2 \sin ^{2} x\right)}$

$=\frac{2(\sin 2 x+\cos 2 x)}{4 \cos x \sin ^{2} x+2 \sin x \cos 2 x}$

$=\frac{2(\sin 2 x+\cos 2 x)}{2 \times 2 \sin x \cos x \sin x+2 \sin x \cos 2 x}$

$=\frac{2(\sin 2 x+\cos 2 x)}{2 \sin 2 x \sin x+2 \sin x \cos 2 x}$

$=\frac{2(\sin 2 x+\cos 2 x)}{2 \sin x(\sin 2 x+\cos 2 x)}$

$=\frac{1}{\sin x}$

$=\operatorname{cosec} x$