The rate of change of the function
$$f ( x ) = - \csc x - \sin x$$
is given by the expression
$$\csc x \cot x - \cos x \text {. }$$
Which of the following is its simplification?
A) $- \csc x \sec ^ { 2 } x$
B) $\tan ^ { 2 } x \cos x$
C) $\cot ^ { 2 } x \cos x$
D) $- \cot x \cos ^ { 2 } x$
E) $\sec ^ { 2 } x$

Question

Quizplus · Accepted Answer

The given expression $\csc x \cot x - \cos x$ simplifies to $\cot ^ { 2 } x \cos x$ by recognizing that $\csc x \cot x = \frac{1}{\sin x} \cdot \frac{\cos x}{\sin x} = \frac{\cos x}{\sin^2 x} = \cot^2 x \cos x$, and the $-\cos x$ term remains unchanged.

The Rate of Change of the Function f(x)=−csc⁡x−sin⁡xf ( x ) = - \csc x - \sin xf(x)=−cscx−sinx

The Rate of Change of the Function $f ( x ) = - \csc x - \sin x$