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** Step-by-step Solution **

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Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\cos\left(x\right)$ and $g=\csc\left(x\right)$

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$\frac{d}{dx}\left(\cos\left(x\right)\right)\csc\left(x\right)+\cos\left(x\right)\frac{d}{dx}\left(\csc\left(x\right)\right)$

Learn how to solve factor problems step by step online. Find the derivative of cos(x)csc(x). Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=\cos\left(x\right) and g=\csc\left(x\right). The derivative of the cosine of a function is equal to minus the sine of the function times the derivative of the function, in other words, if f(x) = \cos(x), then f'(x) = -\sin(x)\cdot D_x(x). Applying the trigonometric identity: \sin\left(\theta\right)\cdot\csc\left(\theta\right)=1. Taking the derivative of cosecant function: \frac{d}{dx}\left(\csc(x)\right)=-\csc(x)\cdot\cot(x)\cdot D_x(x).

** Final Answer

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