What is the correct way to divide a fraction?

Question

This is a very basic question, but i'm struggling with it. Can someone explain the rules when dividing a fraction like this: $$\frac{\cos(\pi x)\sin(\pi x)}{\Large{\frac{\cos(\pi x)}{\sin(\pi x)}}}$$

Is this equal to $\large{\frac{\cos(\pi x)\sin^{2}(\pi x)}{\cos(\pi x)}}$ or $\large{\frac{\cos(\pi x)\sin(\pi x)}{\cos(\pi x)\sin(\pi x)}} = 1$

And why is it equal to one of them but not the other? Also, could you show how one can get from $\large{\frac{\cos(\pi x)\sin(\pi x)}{\Large{\frac{\cos(\pi x)}{\sin(\pi x)}}}}$ to $\tan(\pi x)$?

Answer 1

Let $A = \cos(\pi x)$ and $B = \sin(\pi x)$, then $$\require{cancel}\frac{AB}{\frac{A}{B}} = \frac{AB}{\frac{A}{B}}\cdot\frac{B}{B} = \frac{ABB}{\frac{A}{\cancel{B}}\cancel{B}} = \frac{\bcancel{A}B^2}{\bcancel{A}} = B^2$$ First we multiplied by $1 = B/B$, after that we just simplified.

Therefore we have that $$\frac{\cos(\pi x)\sin(\pi x)}{\frac{\cos(\pi x)}{\sin(\pi x)}} = \sin^2(\pi x) $$ which is equal to the first of your two options.

For your second question, remember that $$\tan (x) = \frac{\sin(x)}{\cos(x)} $$ so to get from $\sin^2(\pi x)$ to $\tan (\pi x)$ we need to divide once by $\sin(\pi x)$ (to remove the square) and once by $\cos (\pi x)$, so in total divide by $\sin(\pi x) \cdot\cos(\pi x)$.

Answer 2

Notice this

$$ \frac{(\frac{a}{b})}{c} = \frac{a}{b}\frac{1}{c} = \frac{a}{bc} $$

but also

$$\frac{a}{(\frac{b}{c})} = \frac{a}{1}\frac{c}{b} = \frac{ac}{b}.$$

In this case we have

$$ \frac{\cos{(\pi x)}\sin{(\pi x)}}{\frac{\cos{(\pi x)}}{\sin{(\pi x)}}} = \frac{\cos{(\pi x)}\sin{(\pi x)}}{1}\frac{\sin{(\pi x)}}{\cos{(\pi x)}} = \sin^2{(\pi x)}.$$.

Answer 3

When dividing by a fraction, multiply by the reciprocal $$\frac{\frac{a}{b}}{\color{red}{\frac{p}{q}}}=\frac{\frac{a}{b}}{\frac{p}{q}}\cdot\frac{\frac{q}{p}}{\frac{q}{p}}=\frac{\frac{a}{b}\cdot\frac{q}{p}}{\frac{pq}{pq}}=\frac{\frac{a}{b}\cdot\frac{q}{p}}{1}= \frac{a}{b}\cdot\color{red}{\frac{q}{p}}$$