1. **State the problem:** Simplify the expression $$\csc^2 x \tan x - \sin x \sec x.$$\n\n2. **Recall definitions and identities:**\n- $$\csc x = \frac{1}{\sin x}$$\n- $$\sec x = \frac{1}{\cos x}$$\n- $$\tan x = \frac{\sin x}{\cos x}$$\n- Pythagorean identity: $$\csc^2 x = 1 + \cot^2 x$$\n\n3. **Rewrite the expression using definitions:**\n$$\csc^2 x \tan x - \sin x \sec x = \left(\frac{1}{\sin^2 x}\right) \left(\frac{\sin x}{\cos x}\right) - \sin x \left(\frac{1}{\cos x}\right)$$\n\n4. **Simplify each term:**\n$$= \frac{1}{\sin^2 x} \cdot \frac{\sin x}{\cos x} - \frac{\sin x}{\cos x} = \frac{1}{\sin x \cos x} - \frac{\sin x}{\cos x}$$\n\n5. **Find common denominator $$\sin x \cos x$$:**\n$$= \frac{1}{\sin x \cos x} - \frac{\sin x}{\cos x} \cdot \frac{\sin x}{\sin x} = \frac{1}{\sin x \cos x} - \frac{\sin^2 x}{\sin x \cos x}$$\n\n6. **Combine the fractions:**\n$$= \frac{1 - \sin^2 x}{\sin x \cos x}$$\n\n7. **Use Pythagorean identity $$1 - \sin^2 x = \cos^2 x$$:**\n$$= \frac{\cos^2 x}{\sin x \cos x}$$\n\n8. **Cancel one $$\cos x$$ factor:**\n$$= \frac{\cancel{\cos^2 x}}{\sin x \cancel{\cos x}} = \frac{\cos x}{\sin x}$$\n\n9. **Recognize the result:**\n$$\frac{\cos x}{\sin x} = \cot x$$\n\n**Final answer:** $$\cot x$$