1. Stating the problem: Simplify and verify the trigonometric identity $$\cos \theta \sqrt{\cot^2 \theta + 1} = \sqrt{\csc^2 \theta - 1}$$. 2. Recall the Pythagorean identities: - $$\cot^2 \theta + 1 = \csc^2 \theta$$ - $$\csc^2 \theta - 1 = \cot^2 \theta$$ 3. Substitute $$\cot^2 \theta + 1$$ with $$\csc^2 \theta$$ inside the square root on the left side: $$\cos \theta \sqrt{\cot^2 \theta + 1} = \cos \theta \sqrt{\csc^2 \theta}$$ 4. Since $$\sqrt{\csc^2 \theta} = |\csc \theta|$$, the left side becomes: $$\cos \theta |\csc \theta|$$ 5. On the right side, substitute $$\csc^2 \theta - 1$$ with $$\cot^2 \theta$$: $$\sqrt{\csc^2 \theta - 1} = \sqrt{\cot^2 \theta} = |\cot \theta|$$ 6. So the equation is now: $$\cos \theta |\csc \theta| = |\cot \theta|$$ 7. Recall that $$\csc \theta = \frac{1}{\sin \theta}$$ and $$\cot \theta = \frac{\cos \theta}{\sin \theta}$$, so: $$\cos \theta |\csc \theta| = \cos \theta \frac{1}{|\sin \theta|} = \frac{\cos \theta}{|\sin \theta|}$$ 8. The right side is: $$|\cot \theta| = \left| \frac{\cos \theta}{\sin \theta} \right| = \frac{|\cos \theta|}{|\sin \theta|}$$ 9. Therefore, the equality holds if $$\cos \theta$$ is non-negative, since: $$\frac{\cos \theta}{|\sin \theta|} = \frac{|\cos \theta|}{|\sin \theta|}$$ 10. Final conclusion: The identity holds true for $$\cos \theta \geq 0$$. Answer: $$\cos \theta \sqrt{\cot^2 \theta + 1} = \sqrt{\csc^2 \theta - 1}$$ is valid when $$\cos \theta \geq 0$$.