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Answer:  The correct option is (c). \(\csc x=\dfrac{1}{\sin x}.\)

Step-by-step explanation:  We are given to select the correct basic identity that we will use to verify the following:

\(\dfrac{\sin x+1}{\sin x}=1+\csc x.\)

We have

\(\dfrac{1+\sin x}{\sin x}\\\\\\=\dfrac{1}{\sin x}+\dfrac{\sin x}{\sin x}\\\\\\=1+\dfrac{1}{\sin x}.\)

In order to verify the given trigonometric equation, we must have

\(1+\dfrac{1}{\sin x}=1+\csc x\\\\\\\Rightarrow \dfrac{1}{\sin x}=\csc x\\\\\\\Rightarrow \csc x=\dfrac{1}{\sin x}.\)

Thus, the required identity that we will use is

\(\csc x=\dfrac{1}{\sin x}.\)

Option (c) is CORRECT.

Answer:

C

Step-by-step explanation:

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