Question

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

Solution

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Answered 2 years ago
Answered 2 years ago
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Use the following properties of the trigonometric functions to obtain:

sinxcsc(x)=(1)sinx(cscx)=(2)sinx1sinx=(3)1\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*}

Explanations\textbf{Explanations}:

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

csc(x)=cscx\csc (-x) =- \csc x

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

cscθ=1sinx\csc \theta = \frac{1}{\sin x}

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

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