1. **Problem:** Prove the identity $\cot x (\cot x + \tan x) = \csc^2 x$. 2. **Recall the definitions and identities:** - $\cot x = \frac{\cos x}{\sin x}$ - $\tan x = \frac{\sin x}{\cos x}$ - $\csc x = \frac{1}{\sin x}$ - Pythagorean identity: $\sin^2 x + \cos^2 x = 1$ 3. **Start with the left-hand side (LHS):** $$\cot x (\cot x + \tan x) = \cot^2 x + \cot x \tan x$$ 4. **Substitute the definitions:** $$= \left(\frac{\cos x}{\sin x}\right)^2 + \frac{\cos x}{\sin x} \cdot \frac{\sin x}{\cos x}$$ 5. **Simplify each term:** $$= \frac{\cos^2 x}{\sin^2 x} + 1$$ 6. **Use the Pythagorean identity:** $$= \frac{\cos^2 x + \sin^2 x}{\sin^2 x} = \frac{1}{\sin^2 x}$$ 7. **Rewrite the right-hand side (RHS):** $$\csc^2 x = \frac{1}{\sin^2 x}$$ 8. **Conclusion:** LHS $= \frac{1}{\sin^2 x} = $ RHS, so the identity is proven. **Final answer:** $\cot x (\cot x + \tan x) = \csc^2 x$ is true.