MWF
Armando Ramirez
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The expression $\frac{\cos x-{\sin }^{2}x}{\cos x}\cdot \csc x$ is to be the left hand side of an equation that is an identity. Which one of the following four expressions can be used as the right hand side of the equation to complete the identity?

Step 1 ：The given expression is $\frac{\cos x-{\sin }^{2}x}{\cos x}\cdot \csc x$.

Step 2 ：We can simplify this expression by using the trigonometric identities. The identity ${\sin }^{2}x+{\cos }^{2}x=1$ can be used to replace ${\sin }^{2}x$ with $1-{\cos }^{2}x$. Also, $\csc x$ is the reciprocal of $\sin x$, so we can replace $\csc x$ with $\frac{1}{\sin x}$.

Step 3 ：After these replacements, we get the expression $\frac{\cos x-(1-{\cos }^{2}x)}{\cos x}\cdot \frac{1}{\sin x}$.

Step 4 ：Simplifying this expression further, we get $-\tan (x)+\frac{1}{\sin (x)}$.

Step 5 ：We know that $\tan (x)=\frac{\sin (x)}{\cos (x)}$ and $\frac{1}{\sin (x)}=\csc (x)$. So, we can replace $\tan (x)$ with $\frac{\sin (x)}{\cos (x)}$ and $\frac{1}{\sin (x)}$ with $\csc (x)$ in the simplified expression.

Step 6 ：The simplified expression is still $-\tan (x)+\frac{1}{\sin (x)}$. It seems that the expression cannot be simplified further.

Step 7 ：Final Answer: The right hand side of the equation that completes the identity is $\boxed{{\displaystyle -\tan (x)+\frac{1}{\sin (x)}}}$.