Select your answer Simplify \( \frac{\cot ^{2}(x) \sin (x)}{\tan (x) \csc (x)} \) \( \cos x \) \( \sin ^{2} x \) \( \frac{\cos ^{3} x}{\sin x} \) \( \cos x \sin x \) \( \frac{\cos ^{2} x}{\sin ^{2} x} \)

To simplify \( \frac{\cot ^{2}(x) \sin (x)}{\tan (x) \csc (x)} \): 1. Rewrite the trigonometric functions in terms of sine and cosine: - \( \cot(x) = \frac{\cos(x)}{\sin(x)} \) - \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) - \( \csc(x) = \frac{1}{\sin(x)} \) 2. Replace the cotangent and tangent: \[\cot^2(x) = \left(\frac{\cos(x)}{\sin(x)}\right)^2 = \frac{\cos^2(x)}{\sin^2(x)}\] \[\tan(x) \csc(x) = \frac{\sin(x)}{\cos(x)} \cdot \frac{1}{\sin(x)} = \frac{1}{\cos(x)}\] 3. Substitute these into the expression: \[ \frac{\cot^2(x) \sin(x)}{\tan(x) \csc(x)} = \frac{\frac{\cos^2(x)}{\sin^2(x)} \sin(x)}{\frac{1}{\cos(x)}} \] This simplifies to: \[ \frac{\cos^2(x) \sin(x)}{\sin^2(x)} \cdot \cos(x) = \frac{\cos^3(x)}{\sin(x)} \] Thus, the final result is \( \frac{\cos^{3}(x)}{\sin(x)} \).