1. The problem is to verify the trigonometric identity: $$\frac{1}{\sin x} + \frac{1}{\cos x} \div \left(\frac{1}{\sin x} - \frac{1}{\cos x}\right) = \frac{\cos^2 x - \sin^2 x}{1 - 2 \cos x \sin x}$$ 2. Start by simplifying the left-hand side (LHS). The expression is a complex fraction: $$\frac{\frac{1}{\sin x} + \frac{1}{\cos x}}{\frac{1}{\sin x} - \frac{1}{\cos x}}$$ 3. Find a common denominator for numerator and denominator separately: Numerator: $$\frac{1}{\sin x} + \frac{1}{\cos x} = \frac{\cos x + \sin x}{\sin x \cos x}$$ Denominator: $$\frac{1}{\sin x} - \frac{1}{\cos x} = \frac{\cos x - \sin x}{\sin x \cos x}$$ 4. Substitute back into the complex fraction: $$\frac{\frac{\cos x + \sin x}{\sin x \cos x}}{\frac{\cos x - \sin x}{\sin x \cos x}} = \frac{\cos x + \sin x}{\cos x - \sin x}$$ 5. Now simplify the right-hand side (RHS): Recall the identity: $$\cos^2 x - \sin^2 x = \cos 2x$$ and $$1 - 2 \cos x \sin x = 1 - \sin 2x$$ So RHS is: $$\frac{\cos 2x}{1 - \sin 2x}$$ 6. Use the tangent half-angle substitution or rewrite numerator and denominator in terms of $\tan x$ to compare with LHS. Alternatively, multiply numerator and denominator of LHS by $\cos x + \sin x$: $$\frac{\cos x + \sin x}{\cos x - \sin x} \times \frac{\cos x + \sin x}{\cos x + \sin x} = \frac{(\cos x + \sin x)^2}{\cos^2 x - \sin^2 x}$$ 7. Expand numerator: $$(\cos x + \sin x)^2 = \cos^2 x + 2 \cos x \sin x + \sin^2 x = 1 + 2 \cos x \sin x$$ So the expression becomes: $$\frac{1 + 2 \cos x \sin x}{\cos^2 x - \sin^2 x}$$ 8. Notice this is the reciprocal of the RHS expression: $$\frac{1 + 2 \cos x \sin x}{\cos^2 x - \sin^2 x} = \frac{1}{\frac{\cos^2 x - \sin^2 x}{1 + 2 \cos x \sin x}}$$ But the RHS given is: $$\frac{\cos^2 x - \sin^2 x}{1 - 2 \cos x \sin x}$$ 9. This suggests the original problem might have a typo or the expressions are not equal as stated. 10. Conclusion: The simplified LHS is: $$\frac{\cos x + \sin x}{\cos x - \sin x}$$ which can be rewritten as: $$\frac{1 + 2 \cos x \sin x}{\cos^2 x - \sin^2 x}$$ and does not equal the RHS expression given. Therefore, the identity as stated is not correct. Final answer: $$\frac{\frac{1}{\sin x} + \frac{1}{\cos x}}{\frac{1}{\sin x} - \frac{1}{\cos x}} = \frac{\cos x + \sin x}{\cos x - \sin x}$$