1. **Stating the problem:** Simplify the expression $$\frac{\sin^2 x - \cos^2 x}{\cos x \sin x}$$ and relate it to the identity $$\frac{\sin^2 x + \cos^2 x}{\cos x \sin x}$$. 2. **Recall important trigonometric identities:** - Pythagorean identity: $$\sin^2 x + \cos^2 x = 1$$ - Difference of squares: $$a^2 - b^2 = (a-b)(a+b)$$ 3. **Simplify the numerator:** $$\sin^2 x - \cos^2 x = (\sin x - \cos x)(\sin x + \cos x)$$ 4. **Rewrite the original expression:** $$\frac{(\sin x - \cos x)(\sin x + \cos x)}{\cos x \sin x}$$ 5. **No direct factor cancellation is possible with denominator, so consider rewriting numerator using cosine double angle identity:** Recall that $$\cos 2x = \cos^2 x - \sin^2 x$$, so $$\sin^2 x - \cos^2 x = - (\cos^2 x - \sin^2 x) = -\cos 2x$$ 6. **Substitute back:** $$\frac{\sin^2 x - \cos^2 x}{\cos x \sin x} = \frac{-\cos 2x}{\cos x \sin x}$$ 7. **Final simplified form:** $$\boxed{\frac{-\cos 2x}{\cos x \sin x}}$$ 8. **Note on the other expression:** $$\frac{\sin^2 x + \cos^2 x}{\cos x \sin x} = \frac{1}{\cos x \sin x}$$ by the Pythagorean identity. This shows the difference between the two expressions clearly.