Multiply trig functions of the same base?

Is $\frac{1}{\sin(x)}\frac{\cos(x)}{\sin(x)} = \frac{\cos(x)}{\sin^2(x)}$ correct?

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Yes, it's correct. –  Daniel Fischer Feb 22 at 23:46

$$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$$ iff $b,d \not= 0$
always holds. With $a=1$, $b=d=\sin(x)$ and $c=\cos(x)$ we get the result:
$$\frac{\cos(x)}{\sin^2(x)}$$