Multiplication of fractions

Hey,

I have a small question about multiplication.

From what I learned at school 1/5/6 equals to 6/5 (I moved 6 to the top)

Is this always true ? How about this example:

It is known that x/tanx equals

x/(cox/sinx) equals

x * 1/(cox/sinx)

Is it legal to say it equals

x * sinx/cosx ? (Move the sin upwards)

Thanks a lot.

Re: Multiplication of fractions

The process you mention is true, however in your example with the tangent function, you would write:

$\displaystyle \frac{x}{\tan(x)}=\frac{x}{\frac{\sin(x)}{\cos(x)} }=\frac{x\cos(x)}{\sin(x)}$

Re: Multiplication of fractions

(x*cosx)/(sinx) is defined in pi /2 (Equals to zero)

According to what your saying, x/tanx is defined in pi/2 .. That is of course not true.

So what is the right answer ? :)

Re: Multiplication of fractions

I should have added for $\displaystyle \cos(x)\ne0$.

I was simply pointing out that $\displaystyle \tan(x)\ne\frac{\cos(x)}{\sin(x)}$ as you stated in your first post.

Re: Multiplication of fractions

Re: Multiplication of fractions

also, while tan(x) is not defined at pi/2, we CAN define 1/tan(x) (another form of cot(x)) at pi/2, as 0 (this makes cot(x) continuous at 0).

this often happens: we can define something that is a/(undefined) sometimes, in a way that "doesn't lead to contradictions". you do have to be careful with this, though.