Question

# Verify the identity. $\sin$ x $\csc$(-x) = -1

Solution

Verified
Step 1
1 of 2

Use the following properties of the trigonometric functions to obtain:

\begin{align*} \sin x \csc (-x)&\overset{{\color{#4257b2}{(1)}}}{=}\sin x(-\csc x) \\ &\overset{{\color{#4257b2}{(2)}}}{=}-\sin x \cdot \frac{1}{\sin x}\\ &\overset{{\color{#4257b2}{(3)}}}{=}-1 \end{align*}

$\textbf{Explanations}$:

$\text{\color{#4257b2}{(1)}}$: Use the $\textit{Negative Angle Identity}$

$\csc (-x) =- \csc x$

$\text{\color{#4257b2}{(2)}}$: Use the $\textit{Reciprocal Identity}$

$\csc \theta = \frac{1}{\sin x}$

$\text{\color{#4257b2}{(3)}}$ : Cancel the like terms.

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