quick question about reciprocal identity

i am dealing with a function involving sqrt x/ csc x. my textbook shows this problem simplifying to (sqrt x sin x). did they use the reciprocal identity? my trig textbook states that sin x= 1/ csc x. is it true to say then that (1)sin x= 1/ csc x? or in this case, sqrt x sin x= sqrt x/ csc x?

Re: quick question about reciprocal identity

By definition, $\displaystyle \csc x = 1 / \sin x$, so $\displaystyle \csc x$ is undefined for $\displaystyle x=\pi k$, $\displaystyle k\in\mathbb{Z}$. Therefore, $\displaystyle \csc x\ne0$ for all x and $\displaystyle \sin x = 1 / \csc x$ for $\displaystyle x\ne\pi k, k\in\mathbb{Z}$.