1. **State the problem:** Verify the identity $$\cot^3 x = \cot x \cdot (\csc^2 x - \cot x)$$. 2. **Recall definitions and identities:** - $$\cot x = \frac{\cos x}{\sin x}$$ - $$\csc x = \frac{1}{\sin x}$$, so $$\csc^2 x = \frac{1}{\sin^2 x}$$ - Pythagorean identity: $$\csc^2 x = 1 + \cot^2 x$$ 3. **Start with the right-hand side (RHS):** $$\cot x \cdot (\csc^2 x - \cot x)$$ Substitute $$\csc^2 x = 1 + \cot^2 x$$: $$= \cot x \cdot (1 + \cot^2 x - \cot x)$$ 4. **Simplify inside the parentheses:** $$1 + \cot^2 x - \cot x = (1 - \cot x) + \cot^2 x$$ 5. **Multiply $$\cot x$$ through:** $$= \cot x \cdot 1 - \cot x \cdot \cot x + \cot x \cdot \cot^2 x = \cot x - \cot^2 x + \cot^3 x$$ 6. **Rearrange terms:** $$= \cot^3 x - \cot^2 x + \cot x$$ 7. **Compare with left-hand side (LHS):** LHS is $$\cot^3 x$$ only. 8. **Check if RHS equals LHS:** RHS has extra terms $$- \cot^2 x + \cot x$$, so the original identity as stated is not true. 9. **Conclusion:** The expression $$\cot^3 x = \cot x \cdot (\csc^2 x - \cot x)$$ is **not** an identity because the RHS expands to $$\cot^3 x - \cot^2 x + \cot x$$ which is not equal to $$\cot^3 x$$ alone. **Final answer:** The given equation is false; it is not an identity.