$$\frac{\sqrt{\operatorname{cosec}x-1}}{\sqrt{\operatorname{cosec}x+1}}$$ is equal to

$$\frac{\sqrt{\operatorname{cosec}x-1}}{\sqrt{\operatorname{cosec}x+1}}=\frac{\sqrt{\operatorname{cosec}x-1}}{\sqrt{\operatorname{cosec}x+1}}\times\frac{\sqrt{\operatorname{cosec}x-1}}{\sqrt{\operatorname{cosec}x-1}}$$

$$=\frac{\operatorname{cosec}x-1}{\sqrt{\operatorname{cosec}^2x-1}}$$

$$=\frac{\operatorname{cosec}x-1}{\sqrt{\cot^2x}}$$

$$=\frac{\operatorname{cosec}x-1}{\cot x}$$

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

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

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

$$=\sec x-\tan x$$

Hence, the correct answer is Option D